$$A^2 + B^2 = (A + B)^2 - 2A^{1/2}BA^{1/2} - [A^{1/2}, [A^{1/2}, B]] \le (A + B)^2 + (2\alpha + \varepsilon)A^2 + 2Ab + b,$$ which establishes (21.4).

Let $\mathcal{A}(C)$ be the graph of the operator $C$. For any hyper infinite sequence $\{C_n\}$, $n \in \mathbb{N}$ of #-densely defined operators we define

$$\mathcal{L}_{*\infty}(C) = \{\varphi, \chi | \varphi = \#\lim_{n \to *\infty} \varphi_n \in D(C_n), \chi = \#\lim_{n \to *\infty} C_n \varphi_n\}. \quad (21.9)$$

In general, $\mathcal{L}_{*\infty}(C)$ will not be the graph of an operator. If the hyper infinite sequence $\{C_n\}$, $n \in \mathbb{N}$ #-converges strongly on a #-dense domain $D$ to an operator $C^*$, namely, $C^*\psi = \#\lim_{n \to *\infty} C_n^*\psi, \psi \in D$, then $\mathcal{L}_{*\infty}$ is the graph of some operator $C^*$. In particular, if each $C_n$ is self #-adjoint, and if the $C_n$ #-converge on a #-dense set $D$ to an operator $C$ is defined on $D$, then $\mathcal{L}_{*\infty} = \mathcal{L}_{*\infty}(C_{*\infty})$ and $C_{*\infty}$ is a symmetric extension of $C$.

**Definition 21.1** G-#-CONVERGENCE. The hyper infinite sequence of operators $\{C_n\}$, $n \in \mathbb{N}$ #-converge strongly to $C_{*\infty}$ in the sense of graphs, written

$$C_n \to_{\#G} C_{*\infty} \qquad (21.10)$$

if $\mathcal{L}_{*\infty}$ is the graph of a #-densely defined operator $C_{*\infty}$.

**Remark 21.1** Note that for a hyper infinite sequence of uniformly bounded in $\mathbb{R}_c^\#$ operators $C_n^*$, $n \in \mathbb{N}$ such that $C_n \to_{\#G} C_{*\infty}$, $C_{*\infty}$ is the usual strong #-limit of the operators $\{C_n\}$, $n \in \mathbb{N}$ and is everywhere defined.

**Definition 21.2.** R-#-CONVERGENCE. Let the resolvents $R_n(z) = (C_n - z)^{-1}$, $n \in \mathbb{N}$ exist for some $z \in \mathbb{R}_c^\#$, and be uniformly hyper bounded in $n$. The operators $C_n$ #-converge strongly to $C_{*\infty}$ in the sense of resolvents, written az

$$C_n \to_{\#R} C_{*\infty} \qquad (21.11)$$

if the resolvents $R_n(z)$ #-converge strongly to an operator $R(z)$, which has a #-densely defined inverse.

**Remark 21.2** Note that in that case, the operator $C_{*\infty} = R^{-1}(z) + z$ exists for all $z \in \mathbb{C}_c^\#$, for which the strong #-limit of the $R_n(z)$ exists, and $R^{-1}(z) + z$ is independent of $z$.

**Remark 21.3** Note that G-#-convergence is weaker than R-#-convergence, in the case $C_n = C_n^*$ at least, because, as we shall show, in this case $C_n \to_{\#R} C_{*\infty}$ implies $C_n \to_{\#G} C_{*\infty}$. It seems likely that

G-#-convergence is strictly weaker than R-#-convergence; this could be established by giving an example for which $C_n^* = C_n \to_{\#G} C_{*\infty}$ with $C_{*\infty}$ not maximal symmetric. The importance of G-#-convergence is that it is technically easier to verify-and gives less information about the #-limit-than R-#-convergence, while automatically selecting the correct domain in the case that R-#-convergence also holds. The most familiar examples of G-#-convergence occur where there is $C_n$ strong #-convergence on a #-dense domain.

A less trivial example occurs where there is $D(C_n)$ is independent of $n \in \mathbb{N}$, but apparently $D(C) \cap D(C_n) = \{0\}$. We have the following connection between G and R-#-convergence for a hyper infinite sequence of #-selfadjoint operators.

**Proposition 21.3** Let $C_n \in \mathbb{N}$ be #-selfadjoint.

(a) The domain $D_{*\infty} = \{\varphi | \{\varphi, \chi\} \in \mathcal{L}_{*\infty}$ for some $\chi\}$ is #-dense in $H$ and only if $C_n \to_{\#G} C_{*\infty}$, and in this case $C_{*\infty}$ is necessarily symmetric.

(b) If $R_n(z) = (C_n - z)^{-1}$, $n \in \mathbb{N}$ #-converges to a bounded in $\mathbb{R}_c^\#$ operator $R(z)$ for an unbounded set of $z$'s with $\|zR(z)\|_\#$ bounded uniformly in $z \in \mathbb{C}_c^\#$, and $n \in \mathbb{N}$ and if $C_n \to_{\#G} C_{*\infty}$, then each $R(z)$ is invertible.

(c) If $R_n(z)$ #-converges to an invertible $R(z)$, then $C_n \to_{\#R} C_{*\infty}$.

(d) If $C_n \to_{\#R} C_{*\infty}$, then $C_n \to_{\#G} C_{*\infty}$, $\mathcal{L}_{*\infty} = \mathcal{L}(C)$, and $C$ is maximal symmetric.

(e) Conversely, if $C_n \to_{\#G} C_{*\infty}$, where $C$ is maximal symmetric, then $C_n \to_{\#R} C_{*\infty}$.

**Proof.** (a) Suppose that $D_{*\infty}$ is #-dense and let $\{0, \psi\} \in \mathcal{L}_{*\infty}$. Then, for some $\theta_n$ in each $D(C_n)$, $\theta_n \to_{\#0} C_n \theta_n \to_{\#\psi}$.
Let $\varphi \in \mathcal{L}_{*\infty}$, $\varphi = \#\lim_{n \to *\infty} \varphi_n$, $\chi = \#\lim_{n \to *\infty} C_n \varphi_n$.

Then, $0 = \langle 0, \psi \rangle_{\#} = \#\lim_{n \to *\infty} \langle \theta_n, C_n \varphi_n \rangle_{\#} = \#\lim_{n \to *\infty} \langle C_n \theta_n, \varphi_n \rangle_{\#} = \langle \psi, \varphi \rangle_{\#}$, so $\psi \in D_{*\infty}^{\perp}$ and $\psi = 0$. Thus $\mathcal{L}_{*\infty}$ is the graph of an operator $C$ with domain $D_{*\infty}$, and since $D_{*\infty}$ #-dense by hypothesis, $C_n \to_{\#G} C_{*\infty}$.

Conversely, if $C_n \to_{\#G} C_{*\infty}$, we have $D_{*\infty} = D(C_{*\infty})$ and, since $C$, is assumed to be #-densely defined, $D_{*\infty}$ is #-dense. The expectation values $\langle \varphi, C_{*\infty} \varphi \rangle_{\#} = \#\lim_{n \to *\infty} \langle \varphi_n, C_n \varphi_n \rangle_{\#}$ are real and so $C_{*\infty}$, is symmetric.

(b) Let $z_1$ and $z_2$ be any two z's for which $\#\lim_{n \to *\infty} R_n(z)$ exists. Then $R(z)$ satisfies the resolvent equation

$R(z_1) - R(z_2) = (z_1 - z_2)R(z_1)R(z)$ from which it follows that $R(z)$ has a range and a null space which are independent of $z$. If the null space is zero, $R(z)$ is invertible, and it is sufficient to show that, for large $|z|$, $\|zR(z)\psi + \psi\|_{\#} < \|\psi\|_{\#}$. Since $D_{*\infty}$ is #-dense, we choose $\varphi \in D_{*\infty}$ so that $\|\psi - \varphi\|_{\#} < \varepsilon \approx 0$.

Then, $\|zR(z)\psi + \psi\|_{\#} \le \|zR(z)(\psi - \varphi)\|_{\#} + \|zR(z)\varphi + \varphi\|_{\#} + \|\psi - \varphi\|_{\#}$. For the $z$ under consideration, $\|zR_n(z)\|_{\#}$ and $\|zR(z)\|_{\#}$ are uniformly bounded in $\mathbb{R}_c^\#$ by a constant $M \in \mathbb{R}_{c^+}^\#$. Thus, $\|zR(z)\psi + \psi\|_{\#} \le (M+1)\varepsilon + \|zR(z)\varphi + \varphi\|_{\#}$. Since $\{\varphi, \chi\} \in \mathcal{L}_{*\infty}$, there exists $\varphi_n \in D(C_n)$ such that $\|\varphi_n - \varphi\|_{\#} \to 0$ and $\|C_n\varphi_n - \chi\|_{\#} \to 0$. Thus $|C_n\{\varphi_n\} - \{\varphi\}|_{\#} \le \|\varphi_n - \varphi\|_{\#} \le |z|(R(z) - R_n(z))\|\varphi\|_{\#} + (M+1)\|\varphi - \varphi_n\|_{\#} + \|(C_n - z)^{-1}C_n\varphi_n\|_{\#}$.

$$ \|zR(z)\varphi - \varphi\|_{\#} \le \|z(R(z) - R_n(z))\varphi\|_{\#} + \|zR_n(z)(\varphi_n - \varphi)\|_{\#} + \|zR_n(z)\varphi_n + \varphi_n\|_{\#} + \\ + \|\varphi_n - \varphi\|_{\#} \le |z|(R(z) - R_n(z))\varphi\|_{\#} + (M+1) \|\varphi - \varphi_n\|_{\#} + \|(C_n - z)^{-1}C_n\varphi_n\|_{\#}. $$

We can choose $|z|$ sufficiently infinite large so that $\|(C-z)^{-1}\|_{\#} \le M|z|$ is infinite small, then the last term above is small, uniformly in $n \in *\mathbb{N}$. With this fixed value of $z$, we choose $n$ large enough so that the first two terms are infinite small, and we conclude that $\|zR(z)\varphi + \varphi\|_{\#}$ is arbitrarily infinite small for infinite large $|z|$ and that the null space of $R(z)$ is zero.

(c) In order to show that $C_n \to_{\#R} C_{*\infty}$, we need only show that $C = R(z)^{-1} + z$ is #-densely defined. We show that $\{\text{range}R(z)\}^\perp \subset \text{null}R(z)$ which implies (c). To prove the inclusion we may suppose that $y = \text{Im}(z) \ne 0$, because $R_n(z) + R(z)$ for $z$ in an #-open subset of the complex plane $\tilde{\mathbb{C}}_c^\#$. If $\psi^\perp\{\text{range}R(z)\}$ and if $\notin$ null$R(z)$, then $\|\psi\|_{\#} < \|iyR(z)\psi + \psi\|_{\#} = \#\lim_{n \to *\infty} \|iyR_n(z)\psi + \psi\|_{\#} \le \|\psi\|_{\#}$, which is a contradiction.

(d) Let $C_n \to_{\#R} C_{*\infty}$, and let $\{\varphi, \theta\} \in \mathcal{L}(C)$. Then, for some $\chi \in H^\#$ and some $z$, $\varphi = R(z)\chi = \#\lim_{n \to *\infty} \varphi_n = \#\lim_{n \to *\infty} R_n(z)\chi$ and $C_n\varphi_n = (C_n-z)\varphi_n + z\varphi_n\chi + \varphi z_n \to_{\#} \chi + z\varphi = C\varphi$. Thus, $L(C) \subset \mathcal{A}(C_{*\infty})$ so $D(C) \subset D_{*\infty}$ and since $D(C)$ is #-dense by assumption, $D_{*\infty}$ is #-dense also. By (a), $C_n \to_{\#G} C_{*\infty}$ and $C_{*\infty}$ is a symmetric extension of $C$. However, $C_{*\infty}$ is #-closed and it has a resolvent $R(z)$; therefore, $C$ has defect indices $(0, n)$ and is maximal symmetric. Thus, $C_{*\infty} = C$. (e) Suppose that $\mathcal{L}_{*\infty} = \mathcal{A}(C_{*\infty})$ is the graph of a maximal symmetric operator. For non-real $z$ we have $\|R_n(z)\|_{\#}$ bounded uniformly in $n \in *\mathbb{N}$. Let $\chi \in H^\#$, and write $\chi$ in the form $\chi = (C_{*\infty} \pm i)\psi$ for some $\psi \in D(C_{*\infty})$. Then, $\psi = \#\lim_{n \to *\infty} \psi_n, C_{*\infty}\psi = \#\lim_{n \to *\infty} C_n\psi_n$ and

$$ \chi = \#\lim_{n \to *\infty} (C_{*\infty} \pm i)\psi_n = \#\lim_{n \to *\infty} \psi_n. \text{ Therefore} $$

$$ \#\lim_{n \to *\infty} R_n(\mp i)\chi = \#\lim_{n \to *\infty} R_n(\mp i)\chi_n = \psi = (C_{*\infty} \pm i)^{-1} \text{ and so } C_n \to_{\#R} C_{*\infty} $$

**Remark 21.4** In case the #-limit of the $C_n, n \in *\mathbb{N}$ is actually #-selfadjoint, there are further connections between $G$ and $R$ #-convergence.

**Theorem 21.4** Let $C_n$ be #-selfadjoint. The following conditions are equivalent; (a) $C_n \to_{\#G} C$, and $C = C^*$. (b) $C_n \to_{\#R} C$, and $C = C^*$. (c) The hyper infinite sequences $R_n(z)$ and $[R_n(z)]^*$, $n \in *\mathbb{N}$ #-converge strongly and $\#\lim_{n \to *\infty} R_n(z)$ is invertible for some $z$. (d) Statement (c) holds for all non-real $z \in \tilde{\mathbb{C}}_c^\#$.

*Proof.* The theorem follows from Proposition 21.3 and sect.24.

Now we give estimates which are sufficient to assure that it $G$ #-convergent sequence of operators is $R$ #-convergent, and that the #-limit is maximal symmetric or #-selfadjoint. In order to measure the rate of
#-convergence, we introduce a #-selfadjoint operator $N \ge I$ and the associated non-Archimedean Hilbert spaces $H_\lambda$ with the scalar product

$$ \langle \psi, \psi \rangle_{\#\lambda} = \langle N^{\lambda/2} \psi, N^{\lambda/2} \psi \rangle_{\#}. \qquad (21.12) $$

By standard identifications we have for $\lambda \ge 0: H_\lambda \subset H_0 \subset H_{-1}$ and $H_0 = H$. If $D: H_\alpha \to H_\beta$ is a #-densely defined, bounded in $\mathbb{R}_c^\#$ operator from $H_\alpha$ to $H_\beta$, we let $\|D\|_{\#\alpha,\beta}$ denote its #-norm. Setting $\|D\|_{\#} = \|D\|_{\#0,0}$ we obtain

$$ \|D\|_{\#\alpha,\beta} = \|N^{\beta/2}DN^{-\alpha/2}\|_{\#}. \qquad (21.13) $$

Let $C_n, n \in \mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators, and consider the following three conditions.

(i) Suppose that $C_n - C_m$ is a #-densely defined, bounded in $\mathbb{R}_c^\#$ operator from $H_\lambda$ to $H_{-\lambda}$, for some $\lambda$, and that as $n, m \to \infty$

$$ \|D\|_{\#\lambda, -\lambda} \to_\# 0 \underset{\mathbb{R}_c^\#}{\sim}. \qquad (21.14) $$

(ii) Suppose that, for some $p$ and for an unbounded set of $z = x + iy \in \tilde{C}_c^\#$ in the sector $|x| \le \text{const} \times |y|$,

$$ \|R_n(z)\|_{\#\mu,\lambda} \le M(z), \qquad (21.15) $$

where the bound $M(z)$ is uniform in $n \in \mathbb{N}$.

(iii) Suppose that, for the above $z$'s,

$$ \|R_n(\bar{z})\|_{\#\mu,\lambda} \le M(z). \qquad (21.16) $$

**Theorem 21.5** Let $C_n, n \in \mathbb{N}$ be a hyper infinite sequence of #-selfadjoint operators with a common domain, such that $C_n \to_{\#G} C_{\infty}$. If the conditions (i) and (ii) mentioned above hold, then $C_n \to_{\#R} C_{\infty}$ and $C$ is maximal symmetric.

**Corollary 21.6** If in addition to the hypothesis of Theorem 21.5, condition (iii) also holds, then $C$ is #-selfadjoint.

**Remark 21.5** (1) If $\mu = 0$ in (ii), then the resolvents #-converge uniformly. (2) If the $C_n$ are uniformly semibounded in $\mathbb{R}_c^\#$ from below, then we may choose the $z$ in condition (ii) to be infinite large negative numbers. In that case the conclusion of the Theorem 21.5 is that $C_n \to_{\#R} C = C^*$.

We consider now a singular perturbation $B$ of a #-selfadjoint operator $A$. We give estimates on $B$ which ensure that the sum $A + B$ is #-selfadjoint.

**Abbreviation 21.1** We abbreviate $A^{-\#}$ instead $-\bar{A}$.

**Definition 21.3** A #-core of an operator $C$ is a domain $D$ contained in $D(C)$ such that $C = (C \upharpoonright D)^{-\#}$.

**Lemma 21.7** Let, $A_n, n \in \mathbb{N}$, $B, B_n, n \in \mathbb{N}$ and $C_n = A + B_n, n \in \mathbb{N}$ be #-selfadjoint operators with a common #-core $D$. Assume the hypotheses of Theorem 21.5 and Corollary 21.6 for $C_n, n \in \mathbb{N}$ and suppose also that, for $\theta \in D$,

$$ \|(A - A_n)\theta\|_{\#} + \|(B - B_n)\theta\|_{\#} \to_{\#} 0 \text{ as } n \to \infty, \qquad (21.17) $$

$$ \|A\theta\|_{\#}^2 + \|B\theta\|_{\#}^2 \le \text{const.} \times \|\theta\|_{\#}^2 + \text{const.} \times \|C_n\theta\|_{\#}^2, \qquad (21.18) $$

with constants independent of $n$. Then $A + B$ is #-selfadjoint and $C_n \to_{\#R} (A + B)$. Proof. Let $\psi_n = R_n(z)\chi$. We have the inequality

$$ \|A_n \psi_n\|_{\#} \le \text{const.} \times \|\psi_n\|_{\#} + \text{const.} \times \|C_n \psi_n\|_{\#} \le \text{const.} \times \|(C_n - z) \psi_n\|_{\#} = \text{const.} \times \|\chi\|_{\#} $$
Thus, $|\langle A\theta, \psi \rangle_{\#}| \le \|A\theta\|_{\#}\|\psi - \psi_n\|_{\#} + \|(A - A_n)\theta\|_{\#} \times \|\psi_n\|_{\#} + \text{const.} \|\theta\|_{\#} \times \|\psi\|_{\#}$ and therefore $|\langle A\theta, \psi \rangle_{\#}| \le \text{const.} \|\theta\|_{\#} \times \|\psi\|_{\#}$ for $\theta \in D$ and $\psi = R(z)\chi$. It follows that $\psi \in D(A \upharpoonright D) = D(A)$ and similarity we obtain that $\psi \in D(B)$. Since $D(C)$ is the range of $R(z)$, we have shown that $D(C) = D(A + B)$. Since $\langle \theta, (A + B)\psi \rangle_{\#} = \langle (A + B)\theta, \psi \rangle_{\#} = \# - \lim_{n \to \infty} \langle C_n\theta, \psi_n \rangle_{\#} = \# - \lim_{n \to \infty} \langle \theta, C_n\psi_n \rangle_{\#} = \langle \theta, C\psi \rangle_{\#}$ we have $C \subset A + B$. Thus $A + B$ is a symmetric extension of the #-selfadjoint operator $C$, and so $A + B = C$.

**Remark 21.6** As hypothesis for our next theorem, our second main result, we assume that $N \le A$ and that $N$ and $A$ commute. Let

$$D^{*\infty}(A) = \bigcap_{n \in \mathbb{N}} D(A^n) \qquad (21.19)$$

the elements of $D^{*\infty}(A)$ are called $C^{*\infty}$ vectors for $A$. Assume that $D^{*\infty}(A)$ is a #-core for the #-selfadjoint operator $B$. Also assume that, as bilinear forms on $D^{*\infty} \times D^{*\infty}$ and for some $\alpha$ and $\varepsilon$ in the indicated ranges,

$$0 \le \alpha N + B + \text{const.}, 0 \le \alpha < 1_{\overline{\mathbb{R}}_{c}^{\#}}/2_{\overline{\mathbb{R}}_{c}^{\#}}, \qquad (21.20)$$

$$0 \le \varepsilon A^2 + \text{const} \times B + [A^{1/2}, [A^{1/2}, B]] + \text{const.}, 2\alpha + \varepsilon < 1_{\overline{\mathbb{R}}_{c}^{\#}}. \qquad (21.21)$$

Let $B$ be a bounded in $\overline{\mathbb{R}}_{c}^{\#}$ operator from $H_v$ to $H_{-v}$ and from $H_\alpha$ to $H_\beta$ for some $\alpha, \beta$ and $\beta > 0$, where $(H_\alpha)$ is defined following Theorem 21.4. If $v \ge 2_{\overline{\mathbb{R}}_{c}^{\#}}$, assume that for all $\varepsilon > 0$

$$0 \le \varepsilon N^{\mu+2} + [N^{(\mu+1)/2}, [N^{(\mu+1)/2}, B]] + \text{const.} \qquad (21.22)$$

as bilinear forms on $D^{*\infty} \times D^{*\infty}$, for some $\mu > v - 2_{\overline{\mathbb{R}}_{c}^{\#}}$.

**Theorem 21.6.** Under the above hypothesis, $A + B$ is #-selfadjoint.

Proof Let $N = \text{Ext} - \int_{1}^{*\infty} \lambda d^\# E_\lambda$, and let $B_n = (E_n B E_n)^{\#-}$, $A_n = A$. $E_n$ leaves $D$ invariant and so the domain of $B_n$ contains $D$. For $\theta \in D$

$$|\langle \theta, B_n \theta \rangle_{\#}| = |\langle E_n \theta, B E_n \theta \rangle_{\#}| \le \|B\|_{\#\lambda, -\lambda} \times \|N^{\nu/2} E_n \theta\|_{\#}^2 \le n^\nu \|B\|_{\#\lambda, -\lambda} \times \|\theta\|_{\#}^2 \qquad (21.23)$$

and so $B_n$ is bounded in $\overline{\mathbb{R}}_{c}^{\#}$ operator and essentially #-selfadjoint on $D$. By Corollary 21.8, $C_n = A + B_n$ is essentially #-selfadjoint on $D$ and $D(C_n) = D(A)$.

Let $D_0 \subset D$ be those vectors with #-compact support relative to the spectral #-measure of the operator $A$. In other words, if $\psi \in D_0$, there exist constants $\alpha$ and $\beta$ such that, for all $\mu \ge 0$, $\|A^\mu \psi\|_{\#}^2 \le \alpha \beta^\mu$. It is clear that the vectors $(C_n + b)D_0$ are #-dense in $H^\#$ and are #-analytic vectors for $N^{\mu/2}$ for any $\mu$. Thus by generalized Nelson's theorem [19], $N^{\mu/2}$ is essentially #-selfadjoint on $(C_n + b)D_0$, and hence $(C_n + b)D_0$ is #-dense in $H_\mu$} every $\mu$. In terms of $H^\#$

$$N^{\mu/2} (C_n + b)D \supset N^{\mu/2} (C_n + b)D_0 \qquad (21.24)$$

is #-dense in $H^\#$ for every $\mu$. For $\lambda > v$ and for $m \le n$

$$\|B_n - B_m\|_{\#\lambda, -\lambda} = \|N^{-\lambda/2} (B_n - B_m) N^{-\lambda/2}\| \le 2 \left\| N^{\frac{\lambda-v}{2}} (I - E_m) \right\|_{\#} \|N^{\nu/2} B N^{\nu/2}\|_{\#} \\ \le 2m^{-(\lambda-v)/2} \|B\|_{\#\lambda, -\lambda}. \qquad (21.25)$$

The inequality (21.21) is preserved under the substitution $B \to B_n$. To see this, we multiply (21.21) by $E_n$ on the left and right and notice that

$$\varepsilon E_n A^2 E_n + \text{const.} \times E_n \le \varepsilon A^2 + \text{const.} \qquad (21.26)$$

Similarly we see that (21.20) is preserved under the substitution $B \to B_n$. Thus the bounds (21.18) follow from Proposition 21.2 applied to the operators $A$ and $B_n$. To prove (21.17), let $\theta \in D$. Then,
$$ \|(B - B_n)\theta\|_{\#} \le (\|I - E_n\|\theta\|_{\#} + \|B(I - E_n)\theta\|_{\#} \le \\ n^{-\beta/2}\|B\|_{\alpha\beta} \times I\|N^{\alpha/2}\theta\|_{\#} + n^{-1}\|B\|_{\alpha\beta} \times \|N^{1+(\alpha/2)}\theta\|_{\#}. $$

We set $\mu = \nu + 2$ and use Lemma 21.7. Because of the uniform lower bound (21.18), we can find infinite large negative numbers $-c$ bounded away from the spectrum of $C_n$. If $\nu < 2_{*\mathbb{R}^\#_c}$, then we set $\mu = 0_{*\mathbb{R}^\#_c}$, $\lambda = 2_{*\mathbb{R}^\#_c}$, and, because of (21.18), we have

$$ \|R_n(-c)\|_{\mu\lambda} \times \|NR_n(-c)\|_{\mu\lambda} \times \|AR_n(-c)\|_{\mu\lambda} \le \text{const.} \quad (21.27) $$

Now we assume $\nu \ge 2$ and we use (21.22) to bound

$$ \|R_n(-c)\|_{\mu\lambda} = \|N^{(\mu+2)/2}R_n(-c)N^{-\mu/2}\|_{\mu\lambda}. \quad (21.28) $$

Since a bounded in $\mathbb{R}^\#_c$ operator is determined by its action on any #-dense domain and since $N^{\mu/2}(C_n + c)$ maps onto a #-dense subset of $H^\#$, it is sufficient to show that

$$ N^{\mu+2} \le \text{const.} \times (C_n + c) N^{\mu}(C_n + c) \quad (21.29) $$

as a bilinear form on $D \times D$. We expand the right side as

$$ (C_n + c)N^{\mu}(C_n + c) \varepsilon^2 N^{\mu+2} + \\ +P + \varepsilon(A - \varepsilon N + B_n + c)N^{\mu+1}\{(A - \varepsilon N + B_n + c)\} + \varepsilon N^{\mu+1}(A - \varepsilon N + B_n + c) \quad (21.30) $$

with $0 \le P$ and so it is sufficient to show that, for some $\varepsilon > 0$, $\varepsilon \approx 0$

$$ 0 \le \varepsilon N^{\mu+2} + 2N^{(\mu+1)/2}(A - \varepsilon N + B_n + c/2) N^{(\mu+1)/2} + [N^{(\mu+1)/2}, [N^{(\mu+1)/2}, B_n]] \quad (21.31) $$

on the domain $D \times D$. In these inequalities, $c$ may be chosen independently of $\varepsilon$ and hence $c$ is an arbitrarily large constant. By (21.22), the sum of the first, third and fourth terms are positive and, by (21.20), the second term is positive. We have verified the hypotheses of Lemma 21.7, and the theorem follows; $A + B$ is # -selfadjoint.

**Definition 21.4** Let $M, N, ...$ be a #-closed linear manifolds of a non-Archimedean Banach space $Z$. We denote by $S_M$ the unit sphere of $M$ (the set of all $u \in M$ with $\|u\|_{\#} = 1$). For any two #-closed linear sub manifolds $M, N$ of $Z$, we set: (1) $\delta(M, N) = \sup_{u \in S_M}\{\text{dist}(u, N), (2) \hat{\delta}(M, N) = \max[\delta(M, N), \delta(N, M)]$.

**Remark 21.7** Note that (1) has no meaning if $M = 0$; in this case we define $\delta(0, N) = 0$ for any $N$. On the other hand $\delta(M, 0) = 1$ if $M \ne 0$, as is seen from the definition, $\delta(M, N)$ can also be characterized as the smallest number $\delta$ such that (3) $\text{dist}(u, N) \le \delta\|u\|_{\#}$ for all $u \in M$.

**Definition 21.5** $\hat{\delta}(M, N)$ will be called the gap between the manifolds $M, N$.

**Lemma 21.8** (1) $\delta(M, N) = 0$ if and only if $M \subset N$. (2) $\hat{\delta}(M, N) = 0$ if and only if $M = N$.

(3) $\hat{\delta}(M, N) = \hat{\delta}(N, M)$. (4) $0 \le \delta(M, N) \le 1$, $0 \le \hat{\delta}(M, N) \le 1$.

Proof. Directly from the definitions.

**Definition 21.6** We set: (1) $(M, N) = \sup_{u \in S_M}\text{dist}(u, S_N)$, (2) $\hat{d}(M, N) = \max[d(M, N), d(N, M)]$. Note that (1) does not make sense if either $M$ or $N$ is $0$. In such cases we set (3) $d(0, N) = 0$ for any $N$; $\hat{d}(M, 0) = 2$ for $M \ne 0$.

**Lemma 21.9** $d$ and $\hat{d}$ satisfy the triangle inequalities: (1) $d(L, N) \le d(L, M) + d(M, N)$,

(2) $\hat{d}(L, N) \le \hat{d}(L, M) + \hat{d}(M, N)$.

Proof. The second inequality follows from the first, which in turn follows directly from the definition.

**Definition 21.7** We say that hyper infinite sequence $T_n, n \in \mathbb{N}$ #-converges to operator $T$ ($T_n \to_{g\#} T$) in the generalized sense if $\hat{\delta}(T_n, T) \to_{\#} 0$.

**Theorem 21.7** Let $T: H \to H$ be a #-selfadjoint operator. Then there is a $\delta > 0, \delta \approx 0$ such that any #-closed symmetric operator $S$ with $\hat{\delta}(S, T) < \delta$ is #-selfadjoint, where $\hat{\delta}(S, T)$ denotes the gap between $S$ and $T$.
**Corollary 21.8** Let $T, T_n$ be #-closed symmetric operators and let $\{T_n\}, n \in \mathbb{N}$ #-converge to $T$ in the generalized sense, see Definition 21.7. If $T$ is #-selfadjoint, then $T_n$ is #-selfadjoint for sufficiently large $n \in \mathbb{N} \setminus \mathbb{N}$.

**Corollary 21.9** With the hypothesis of Theorem 21.6, $A + \gamma B \to_{\#R} A$ as $y \to_{\#} 0$. Proof. This is a special case of Corollary 10(a)

**Corollary 21.10** Let $B, B_j$ be singular perturbations of $A$, each satisfying the hypothesis of Theorem 21.6, with constants independent of $j$.

(a) If $B_j - B_l$ is a #-densely defined bounded in $\mathbb{R}_c^\#$ operator from $H_\lambda$ to $H_{-\lambda}$ for a sufficiently large $\lambda$, if $\|B_j - B_l\|_{\#\lambda, -\lambda} \to_\# 0$ and $B_j \to_{\#G} B$, then

$$A + B_j \to_{\#R} A + B. \qquad (21.32)$$

(b) If (a) is true for $\lambda \le 2$ and if there is a #-core $D$ of $C$ with $\subset \cap_j D(B_j)$, then the resolvents of $A + B_j$ #-converge in #-norm to the resolvent of $A + B$.

## 22. Construction of $\mathbb{R}_c^\#$

**Definition 22.1** Let $a_n, n \in \mathbb{N}$ be $\mathbb{R}_c^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_c^\#$ such that:

(i) there is $M \in \mathbb{N}$ such that $\{a_n\}_{n=M}^\infty$ is monotonically decreasing $\mathbb{R}_{cfin}^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_{cfin}^\#$ $\setminus \{0_{\mathbb{R}_c^\#}\}$, We denote these sequences by $\{a_n\}_{n=0}^\infty, \{b_n\}_{n=0}^\infty$, etc.

(ii) there is $N \in \mathbb{N}$ such that for all $n > N$, $a_n \neq 0_{\mathbb{R}_c^\#}$,

(iii) for all $n \in \mathbb{N}$, $a_n \approx 0_{\mathbb{R}_c^\#}$ and for any $\varepsilon$ such that $\varepsilon > 0$, $\varepsilon \approx 0_{\mathbb{R}_c^\#}$ there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n < \varepsilon$ and we denote a set of the all these countable sequences by $\Delta_\omega^{+\downarrow 0}$.

**Definition 22.2** (i) We define a set $\Delta_\omega^{-\downarrow 0}$ by $a_n \in \Delta_\omega^{-\downarrow 0} \Leftrightarrow -a_n \in \Delta_\omega^{+\downarrow 0}$. Note that $\Delta_\omega^{-\downarrow 0} = -\Delta_\omega^{+\downarrow 0}$.

(ii) We define a set $\Delta_\omega^{(-1)\downarrow 0}$ by $a_n \in \Delta_\omega^{(-1)\downarrow 0} \Leftrightarrow a_n^{-1} \in \Delta_\omega^{+\downarrow 0}$. Note that $\Delta_\omega^{(-1)\downarrow 0} = (\Delta_\omega^{+\downarrow 0})^{-1}$.

**Definition 22.3** Let $a_n, n \in \mathbb{N}$ be $\mathbb{R}_c^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_c^\#$ such that:

(i) there is $M \in \mathbb{N}$ such that $\{a_n\}_{n=M}^\infty$ is monotonically decreasing $\mathbb{R}_c^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_{c+\approx}^\#$ $\setminus \{0_{\mathbb{R}_c^\#}\}$,

(ii) there is $N \in \mathbb{N}$ such that for all $n > N$, $a_n \neq 0_{\mathbb{R}_c^\#}$,

(iii) we denote a set of the all these countable sequences by $\Delta_\omega^{+\downarrow 0}$.

**Definition 22.4** (i) We define a set $\Delta_\omega^{-\downarrow 0}$ by $a_n \in \Delta_\omega^{-\downarrow 0} \Leftrightarrow -a_n \in \Delta_\omega^{+\downarrow 0}$. Note that: $\Delta_\omega^{-\downarrow 0} = -\Delta_\omega^{+\downarrow 0}$.

(ii) We define a set $\Delta_\omega^{(-1)\downarrow 0}$ by $a_n \in \Delta_\omega^{(-1)\downarrow 0} \Leftrightarrow a_n^{-1} \in \Delta_\omega^{+\downarrow 0}$. Note that: $\Delta_\omega^{(-1)\downarrow 0} = (\Delta_\omega^{+\downarrow 0})^{-1}$.

**Definition 22.5** Let $a_n, n \in \mathbb{N}$ be $\mathbb{R}_c^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_c^\#$ $\setminus \{0_{\mathbb{R}_c^\#}\}$ such that:

(i) there is $M \in \mathbb{N}$ such that $\{a_n\}_{n=M}^\infty$ is monotonically increasing $\mathbb{R}_{cfin}^\#$- valued countable sequence $a: \mathbb{N} \to \mathbb{R}_{cfin}^\#$ $\setminus \{0_{\mathbb{R}_c^\#}\}$, We denote these sequences by $\{a_n\}_{n=0}^\infty, \{b_n\}_{n=0}^\infty$, etc.

(ii) there is $N \in \mathbb{N}$ such that for all $n > N$, $a_n \neq 0_{\mathbb{R}_c^\#}$,

(iii) for all $n \in \mathbb{N}$, $a_n \approx 0_{\mathbb{R}_c^\#}$ and for any $\varepsilon$ such that $\varepsilon > 0$, $\varepsilon \in \mathbb{R}_{cfin}^\#$ there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n > \varepsilon$ and we denote a set of the all these countable sequences by $\Delta_\omega^{+\downarrow 0}$.

**Definition 22.6** (i) We define a set $\Delta_\omega^{-\downarrow 0}$ by $a_n \in \Delta_\omega^{-\downarrow 0} \Leftrightarrow -a_n \in \Delta_\omega^{+\downarrow 0}$. Note that $\Delta_\omega^{-\downarrow 0} = -\Delta_\omega^{+\downarrow 0}$.

(ii) We define a set $\Delta_\omega^{(-1)\downarrow 0}$ by $a_n \in \Delta_\omega^{(-1)\downarrow 0} \Leftrightarrow a_n^{-1} \in \Delta_\omega^{+\downarrow 0}$. Note that $\Delta_\omega^{(-1)\downarrow 0} = \Delta_\omega^{+\downarrow 0}$.

**Definition 22.7** (i) We define the ordering relation $(\cdot \le \cdot)$ on a set $\Delta_\omega^{+\downarrow 0} \times \Delta_\omega^{+\downarrow 0}$ by: let $\{a_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ and $\{b_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ then $\{a_n\}_{n=0}^\infty \le \{b_n\}_{n=0}^\infty$ iff there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n \le b_n$.

(ii) We define the ordering relation $(\cdot \le \cdot)$ on a set $\Delta_\omega^{-\downarrow 0} \times \Delta_\omega^{-\downarrow 0}$ by: let $\{a_n\}_{n=0}^\infty \in \Delta_\omega^{-\downarrow 0}$ and $\{b_n\}_{n=0}^\infty \in \Delta_\omega^{-\downarrow 0}$, then $\{a_n\}_{n=0}^\infty \le \{b_n\}_{n=0}^\infty$ iff there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n \le b_n$.

(iii) We define the ordering relation $(\cdot \le \cdot)$ on a set $\Delta_\omega^{+\downarrow 0} \times \Delta_\omega^{+\downarrow 0}$ by: let $\{a_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ and $\{b_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ then $\{a_n\}_{n=0}^\infty \le \{b_n\}_{n=0}^\infty$ iff there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n \le b_n$.

**Definition 22.8** (i) We define the ordering relation $(\cdot \le \cdot)$ on a set $\Delta_\omega^{+\downarrow 0} \times \mathbb{R}_{c+}^\#$ by: let $\{a_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ and $x \in \mathbb{R}_{c+}^\#$, then $\{a_n\}_{n=0}^\infty \le x$ iff there is $N \in \mathbb{N}$ such that for all $n > N$: $a_n < x$.

(ii) We define the ordering relation $(\cdot \le \cdot)$ on a set $\Delta_\omega^{+\downarrow 0} \times \mathbb{R}_{c\approx}^\#$ by: let $\{a_n\}_{n=0}^\infty \in \Delta_\omega^{+\downarrow 0}$ and $\varepsilon \in \mathbb{R}_{c\approx}^\#$, then
$\{a_n\}_{n=0}^{\infty} \le \varepsilon$ iff there is $N \in \mathbb{N}$ such that for all $n > N: a_n < \varepsilon$.

(iii) We define the ordering relation ($\leq$) on a set $*\mathbb{R}_{c\approx+}^{\#} \times \Delta_{\omega\approx}^{+10}$ by: let $\{a_n\}_{n=0}^{\infty} \in \Delta_{\omega\approx}^{+10}$ and $\varepsilon \in *\mathbb{R}_{c\approx+}^{\#}$, then
$\varepsilon \le \{a_n\}_{n=0}^{\infty}$ iff there is $N \in \mathbb{N}$ such that for all $n > N: \varepsilon < a_n$.

**Proposition 22.1** Let $\{a_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10}$ and $\{b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10}$, then

(i) $\{a_n\}_{n=0}^{\infty} + \{b_n\}_{n=0}^{\infty} \triangleq \{a_n + b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10}$;

(ii) $\{a_n\}_{n=0}^{\infty} - \{b_n\}_{n=0}^{\infty} \triangleq \{a_n - b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10} \cup \Delta_{\omega}^{-10} \cup \{0_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$, where $\{0_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$ is a countable $0_{*\mathbb{R}_c^{\#}}$-valued sequence;

(iii) $\{a_n\}_{n=0}^{\infty} \times \{b_n\}_{n=0}^{\infty} \triangleq \{a_n \times b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10}$;

Proof. Immediately from definitions.

Proposition 22.2 Let $\{a_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10}$ and $\{b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{(-1)10}$ then

$\{a_n\}_{n=0}^{\infty} \times \{b_n\}_{n=0}^{\infty} \triangleq \{a_n \times b_n\}_{n=0}^{\infty} \in \Delta_{\omega}^{+10} \cup \Delta_{\omega}^{(-1)10} \cup \Delta_{\omega}^{+1\infty} \cup \{1_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$, where $\{1_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$ is a countable $1_{*\mathbb{R}_c^{\#}}$-valued sequence;

Proof. Immediately from definitions.

**Definition 22.9** (i) Let $\{a_n\}_{n=0}^{\infty} \in \mathcal{M}_{\infty} = \Delta_{\omega}^{+10} \cup \Delta_{\omega}^{-10} \cup \Delta_{\omega}^{+1\infty} \Delta_{\omega}^{-1\infty} \cup \{0_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty} \cup \{1_{*\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$, and let $\{A_n\}_{n=0}^{*\infty} \triangleq \overline{\{a_n\}_{n=0}^{*\infty}}$ be a hyper infinite sequence

$$ \{A_n\}_{n=0}^{*\infty} \triangleq \overline{\{a_n\}_{n=0}^{*\infty}} = (a_0, a_1, \dots, a_n, \dots, \{a_n\}_{n=0}^{\infty}, \{a_n\}_{n=0}^{\infty}, \dots), \quad (22.1) $$

i.e. for any infinite $m \in *\mathbb{N} \setminus \mathbb{N}$, $A_m = \{a_n\}_{n=0}^{\infty}$.

(ii) we define a set $\tilde{\mathcal{M}}_{\infty}$ by

$$ \{A_n\}_{n=0}^{*\infty} \in \tilde{\mathcal{M}}_{\infty} \Leftrightarrow (\{A_n\}_{n=0}^{*\infty} = \overline{\{a_n\}_{n=0}^{*\infty}} \wedge (\{a_n\}_{n=0}^{\infty} \in \mathcal{M}_{\infty})) \quad (22.2) $$

**Definition 22.10** (i) Let $\Psi: *\mathbb{N} \to \mathcal{M}_{\infty} \cup (*\mathbb{R}_c^{\#} *^{\mathbb{N}})$ be a hyper infinite sequence and we denote these hyper infinite sequences by $\{\Psi_n\}_{n=0}^{*\infty}$, $\{\Phi_n\}_{n=0}^{*\infty}$, etc.,

(ii) we denote a set of the all these hyper infinite sequences by $\mathcal{R}_{*\infty}$. Note that $\tilde{\mathcal{M}}_{\infty} \subset \mathcal{R}_{*\infty}$.

(iii) Let $\wp: *\mathbb{N} \to \mathcal{M}_{\infty}$ be a hyperfinite sequence and we denote these hyperfinite sequences by $\{\mathcal{F}_n\}_{n=0}^{n=m}, m \in *\mathbb{N} \setminus \mathbb{N}, \{\wp_n\}_{n=0}^{n=k}, k \in *\mathbb{N} \setminus \mathbb{N}, \text{etc.}$,

(iii) Let $\{\mathcal{F}_n\}_{n=0}^{n=m}$ be $\{\mathcal{F}_n\}_{n=0}^{n=m}, m \in *\mathbb{N} \setminus \mathbb{N}$, we define hyper infinite sequence $\{\tilde{\mathcal{F}}_n\}_{n=0}^{n=m}$ by: for $n \le m$, $\overline{\{\mathcal{F}_n\}_{n=0}^{n=m}} = \{\mathcal{F}_n\}_{n=0}^{n=m}$, and for $n > m$, $\mathcal{F}_n = m$.

(iv) We denote a set of the all these hyper infinite sequences by $\mathcal{GF}_{*\infty}$.

(v) Let $\psi \in *\mathbb{R}_c^{\#} *^{\mathbb{N}}$ be a hyper infinite sequence and we denote these hyper infinite sequences by $\{\psi_n\}_{n=0}^{*\infty}$, $\{\phi_n\}_{n=0}^{*\infty}$, $\{\theta_n\}_{n=0}^{*\infty}$, etc.

(vi) Let $\psi \in *\mathbb{R}_c^{\#} *^{\mathbb{N}}$ be a hyper infinite sequence. Assume that there is exists $N \in *\mathbb{N}$ such that $\psi_n \ne 0_{*\mathbb{R}_c^{\#}}$ for $n > N$. Define hyper infinite sequence $\{\phi_n\}_{n=0}^{*\infty}$ of hyperreal numbers from $*\mathbb{R}_c^{\#}$ as follows: for $n \le N$, $\phi_n = 0_{*\mathbb{R}_c^{\#}}$, and for $n > N$, $\phi_n = \frac{1_{*\mathbb{R}_c^{\#}}}{\psi_n}$

$$ \{\phi_n\}_{n=0}^{*\infty} = (0_{*\mathbb{R}_c^{\#}}, 0_{*\mathbb{R}_c^{\#}}, \dots, 0_{*\mathbb{R}_c^{\#}}, \frac{1_{*\mathbb{R}_c^{\#}}}{\psi_{N+1}}, \frac{1_{*\mathbb{R}_c^{\#}}}{\psi_{N+2}}, \dots). \quad (22.3) $$

This definition (22.3) makes sense since, for $n > N$, an is a nonzero hyperreal number, so $1_{*\mathbb{R}_c^{\#}}/\psi_n$ exists. Then $\psi_n \times \phi_n$ is equal to $\psi_n \times 0_{*\mathbb{R}_c^{\#}} = 0_{*\mathbb{R}_c^{\#}}$ for $n \le N$, and equals $\psi_n \times \phi_n = \psi_n \times 1_{*\mathbb{R}_c^{\#}}/\psi_n = 1_{*\mathbb{R}_c^{\#}}$ for $n > N$.
Thus hyper infinite sequence $\{\psi_n\}_{n=0}^{*\infty}$ is invertible in the following sense

$$ (\{\psi_n\}_{n=0}^{*\infty}) \times (\{\phi_n\}_{n=0}^{*\infty}) = (0_{*\mathbb{R}_c^{\#}}, 0_{*\mathbb{R}_c^{\#}}, \dots, 0_{*\mathbb{R}_c^{\#}}, 1_{*\mathbb{R}_c^{\#}}, 1_{*\mathbb{R}_c^{\#}}, \dots). \quad (22.4) $$
If the equality (22.4) holds we say that hyper infinite sequence {$\psi_n$}$_{n=0}^{\infty}$ multiplicative invertible and {$\phi_n$}$_{n=0}^{\infty}$
is multiplicative inverse or reciprocal for hyper infinite sequence {$\psi_n$}$_{n=0}^{\infty}$ and denote it by

$$
(\{\psi_n\}_{n=0}^{\infty})^{-1 \cdot \mathbb{R}_c^{\#}} \triangleq \{\phi_n\}_{n=0}^{\infty}. \qquad (22.5)
$$

Note that

$$
\left( \left( \{\psi_n\}_{n=0}^{\infty} \right)^{-1 \cdot \mathbb{R}_c^{\#}} \right)^{-1 \cdot \mathbb{R}_c^{\#}} \equiv \{\psi_n\}_{n=0}^{\infty}. \qquad (22.6)
$$

**Definition 22.11** Let {$\psi_n$}$_{n=0}^{\infty} \in \mathbb{R}_c^{\#}$ be a hyper infinite sequence such that {$\psi_n$}$_{n=0}^{\infty} \not\equiv \{0_{\mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$ and {$\psi_n$}$_{n=0}^{\infty}$ is not multiplicative invertible. This meant there is exists hyper infinite subsequence {$\psi_{n_i}$}$_{i=0}^{\infty}$ such that $\psi_{n_i} \neq 0_{\mathbb{R}_c^{\#}}$, $i \in \mathbb{N}$ and $\psi_n \equiv 0_{\mathbb{R}_c^{\#}}$ if $\psi_n \notin \{\psi_{n_i}\}_{i=0}^{\infty}$. Define now hyper infinite sequence {$\phi_n$}$_{n=0}^{\infty}$ of hyperreal numbers from $\mathbb{R}_c^{\#}$ as follows: for $n = n_i$, $\phi_n = \frac{1}{\psi_{n_i}}$ and $\phi_n \equiv 0_{\mathbb{R}_c^{\#}}$ if $\psi_n \notin \{\psi_{n_i}\}_{i=0}^{\infty}$. Thus hyper infinite sequence {$\psi_n$}$_{n=0}^{\infty}$ is invertible in the following sense

$$
\{\psi_n\}_{n=0}^{\infty} \times \{\phi_n\}_{n=0}^{\infty} = \{\omega_n\}_{n=0}^{\infty}, \qquad (22.7)
$$

where $\omega_n \equiv 1_{\mathbb{R}_c^\#}$ if $n = n_i$ and $\omega_n \equiv 0_{\mathbb{R}_c^\#}$ if $n \neq n_i$, $i \in \mathbb{N}$. If the equality (22.7) holds we say that hyper infinite sequence {$\psi_n$}$_{n=0}^{\infty}$ multiplicative semi-invertible and {$\phi_n$}$_{n=0}^{\infty}$ is multiplicative semi-inverse or semi-reciprocal for hyper infinite sequence {$\psi_n$}$_{n=0}^{\infty}$ and denote it by

$$
(\{\psi_n\}_{n=0}^{\infty})^{-1 \cdot \mathbb{R}_c^{\#}} \triangleq \{\phi_n\}_{n=0}^{\infty}. \qquad (22.8)
$$

Note that

$$
\left( \left( \{\psi_n\}_{n=0}^{\infty} \right)^{-1 \cdot \mathbb{R}_c^{\#}} \right)^{-1 \cdot \mathbb{R}_c^{\#}} \equiv \{\psi_n\}_{n=0}^{\infty}. \qquad (22.9)
$$

**Definition 22.12** (i) Let {$a_n$}$_{n=0}^{\infty}$, {$b_n$}$_{n=0}^{\infty} \in \Delta_{\omega}^{+10} \cup \Delta_{\omega}^{-10} \cup \Delta_{\omega}^{+1\infty} \Delta_{\omega}^{-1\infty} \cup \{0_{\cdot \mathbb{R}_c^{\#}}\}_{n=0}^{\infty} \cup \{1_{\cdot \mathbb{R}_c^{\#}}\}_{n=0}^{\infty}$ and $x, y \in \mathbb{R}_c^{\#}$, then we define

$$
\{a_n\}_{n=0}^{\infty} \pm \{b_n\}_{n=0}^{\infty} \triangleq \{a_n \pm b_n\}_{n=0}^{\infty}, \qquad (22.10)
$$

$$
x \pm y \{a_n\}_{n=0}^{\infty} \triangleq \{x \pm ya_n\}_{n=0}^{\infty}. \qquad (22.11)
$$

(ii) Let {$\Psi_n$}$_{n=0}^{\infty}$, {$\Phi_n$}$_{n=0}^{\infty} \in \mathbb{R}_{\infty}^*$, and $x, y \in \mathbb{R}_c^*$, then we define

$$
\{\Psi_n\}_{n=0}^{\infty} \pm \{\Phi_n\}_{n=0}^{\infty} \triangleq \{\Psi_n \pm \Phi_n\}_{n=0}^{\infty}, \qquad (22.12)
$$

$$
\{\Psi_n\}_{n=0}^{\infty} \times \{\Phi_n\}_{n=0}^{\infty} \triangleq \{\Psi_n \times \Phi_n\}_{n=0}^{\infty}, \qquad (22.13)
$$

$$
x \pm y \{\Psi_n\}_{n=0}^{\infty} \triangleq \{x \pm y \Psi_n\}_{n=0}^{\infty}. \qquad (22.14)
$$

**Definition 22.13** Let hyper infinite sequence {$\Psi_n$}$_{n=0}^{\infty}$ be in $\mathbb{R}_{\infty}^*$, i.e. for all $n \in \mathbb{N}$, $\Psi_n \in \mathcal{M}_{\infty} \cup (\mathbb{R}_c^{\#})^*$.
Say {$\Psi_n$}$_{n=0}^{\infty}$ #-tends to $0_{\mathbb{R}_c^{\#}} \triangleq 0_{\mathbb{R}_c^{\#}}$, as $n \to \infty$. iff for any given $\varepsilon > 0_{\mathbb{R}_c^{\#}}$, $\varepsilon \approx 0_{\mathbb{R}_c^{\#}}$ there is a hypernatural number $N \in \mathbb{N} \setminus \mathbb{N}$, $N = N(\varepsilon)$ such that for any $n > N$, $|\Psi_n| < \varepsilon$.

**Definition 22.14** Let {$\Psi_n$}$_{n=0}^{\infty}$ be a hyper infinite sequence such that for all $n \in \mathbb{N}$, $\Psi_n \in \mathcal{M}_{\infty}$. We call {$\Psi_n$}$_{n=0}^{\infty}$ a
Cauchy hyper infinite sequence if the difference between its terms #-tends to $0_{\mathbb{R}_c^\#}$. To be precise: given any $\varepsilon > 0_{\mathbb{R}_c^\#}$, $\varepsilon \approx 0_{\mathbb{R}_c^\#}$ there is a hypernatural number $N \in *\mathbb{N}\backslash\mathbb{N}$, $N = N(\varepsilon)$ such that for any $m, n > N$, $|\Psi_n - \Psi_m| < \varepsilon$.

**Theorem 22.1** Let {$\Psi_n$}$_{n=0}^{\infty}$ be in $\mathbb{R}_{\infty}$. If {$\Psi_n$}$_{n=0}^{\infty}$ is a #-convergent hyper infinite sequence (that is, $\Psi_n \to_{\#} \Phi$ as $n \to \infty$ for some $\Phi \in \mathbb{R}_{\infty}$), then {$\Psi_n$}$_{n=0}^{\infty}$ is a Cauchy hyper infinite sequence.

Proof. We know that $\Psi_n \to_{\#} \Phi$. Here is a ubiquitous trick: instead of using $\varepsilon \approx 0_{\mathbb{R}_c^\#}$ in the definition, start with an arbitrary infinitesimal $\varepsilon > 0$, $\varepsilon \approx 0_{\mathbb{R}_c^\#}$ and then choose $N \in *\mathbb{N}\backslash\mathbb{N}$ so that $|\Psi_n - \Phi| < \varepsilon/2$ when $n > N$. Then if $m, n > N$, we have $|\Psi_n - \Psi_m| = |(\Psi_n - \Phi) - (\Psi_m - \Phi)| \le |\Psi_n - \Phi| + |\Psi_m - \Phi| < \varepsilon/2 + \varepsilon/2 = \varepsilon$. This shows that {$\Psi_n$}$_{n=0}^{\infty}$ is a Cauchy hyper infinite sequence.

**Theorem 22.2** If {$\Psi_n$}$_{n=0}^{\infty}$ is a Cauchy hyper infinite sequence, then it is bounded in $*\mathbb{R}_c^\#$; that is, there is some number $M \in *\mathbb{R}_c^\#$ such that $|\Psi_n| \le M$ for all $n \in *\mathbb{N}$.

Proof. Since {$\Psi_n$}$_{n=0}^{\infty}$ is Cauchy, setting $\varepsilon = 1$ we know that there is some $N \in *\mathbb{N}$ such that $|\Psi_n - \Psi_m| < 1$ whenever $m, n > N$. Thus, $|\Psi_{N+1} - \Psi_n| < 1$ for $N > 1$. We can rewrite this as $\Psi_{N+1} - 1 < \Psi_n < \Psi_{N+1} + 1$. This means that $|\Psi_n|$ is less than the maximum of $|\Psi_{N+1} - 1|$ and $|\Psi_{N+1} + 1|$. So, set $M \in *\mathbb{R}_c^\#$ larger than any number in the following hyperfinite list: $\{|\Psi_0|, |\Psi_1|, \dots, |\Psi_N|, |\Psi_{N+1} - 1|, |\Psi_{N+1} + 1|\}$. Then for any term $\Psi_n$, if $n \le N$, then $|\Psi_n|$ appears in the list and so $|\Psi_n| \le M$; if $n > N$, then (as shown above) $||\Psi_n|$ is less than at least one of the last two entries in the list, and so $|\Psi_n| \le M$. Hence, $M$ is a bound for the sequence.

**Definition 22.15** Let $S$ be a set of objects. A relation among pairs of elements of $S$ is said to be an equivalence relation if the following three properties hold: (1) Reflexivity: for any $s \in S$, $s$ is related to $s$. (2) Symmetry: for any $s, t \in S$, if $s$ is related to $t$ then $t$ is related to $s$. (3) Transitivity: for any $s, t, r \in S$, if $s$ is related to $t$ and $t$ is related to $r$, then $s$ is related to $r$.

**Remark 22.1** The following well known proposition goes most of the way to showing that an equivalence relation divides a set into bins.

**Proposition 22.3** Let $S$ be a set, with an equivalence relation on pairs of elements. For $s \in S$, denote by $[s]$ the set of all elements in $S$ that are related to $s$. Then for any $s, t \in S$, either $[s]=[t]$ or $[s]$ and $[t]$ are disjoint. The sets $[s]$ for $s \in S$ are called the equivalence classes, and they are the bins.

Corollary 22.1 If $S$ is a set with an equivalence relation on pairs of elements, then the equivalence classes are non-empty disjoint sets whose union is all of $S$.

**Definition 22.16** Let {$\Psi_{1,n}$}$_{n=0}^{\infty}$ and {$\Psi_{2,n}$}$_{n=0}^{\infty}$ be in $\mathbb{R}_{\infty}$. Say they are equivalent (i.e. related) if $|\Psi_{1,n} - \Psi_{2,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$ as $n \to \infty$.

**Proposition 22.4** Definition 22.15 yields an equivalence relation on $\mathbb{R}_{\infty}$.

Proof. We need to show that this relation is reflexive, symmetric, and transitive. (1) Reflexive: $\Psi_{1,n} - \Psi_{2,n} = 0_{\mathbb{R}_c^\#}$, $n \in *\mathbb{N}$, and the hyper infinite sequence all of whose terms are $0_{\mathbb{R}_c^\#}$ clearly #-converges to $0_{\mathbb{R}_c^\#}$-valued hyper infinite sequence So {$\Psi_{1,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{2,n}$}$_{n=0}^{\infty}$. (2) Symmetric: Suppose that {$\Psi_{1,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{2,n}$}$_{n=0}^{\infty}$, so $|\Psi_{1,n} - \Psi_{2,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$, as $n \to \infty$. But $\Psi_{2,n} - \Psi_{1,n} = -(\Psi_{1,n} - \Psi_{2,n})$, and since only the absolute value $|\Psi_{1,n} - \Psi_{2,n}| = |\Psi_{2,n} - \Psi_{1,n}|$ comes into play in Definition 22.15 it follows that $|\Psi_{2,n} - \Psi_{1,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$, as $n \to \infty$ as well. Hence, {$\Psi_{2,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{1,n}$}$_{n=0}^{\infty}$. (3) Transitive: Suppose {$\Psi_{1,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{2,n}$}$_{n=0}^{\infty}$, and {$\Psi_{2,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{3,n}$}$_{n=0}^{\infty}$. This means that $|\Psi_{1,n} - \Psi_{2,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$ and $|\Psi_{2,n} - \Psi_{3,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$, as $n \to \infty$. To be fully precise, let us fix infinite small $\in *\mathbb{R}_c^\#$; then there exists an $N \in *\mathbb{N}\backslash\mathbb{N}$ such that for all $n > N$, $|\Psi_{1,n} - \Psi_{2,n}| < \varepsilon/2$; also, there exists an $M \in *\mathbb{N}\backslash\mathbb{N}$ such that for all $n > M$, $|\Psi_{2,n} - \Psi_{3,n}| < \varepsilon/2$. Well, then, as long as $> \max(N, M)$, we have that

$$|\Psi_{2,n} - \Psi_{3,n}| = |(\Psi_{1,n} - \Psi_{2,n}) + (\Psi_{2,n} - \Psi_{3,n})| \le |\Psi_{1,n} - \Psi_{2,n}| + |\Psi_{2,n} - \Psi_{3,n}| < \varepsilon/2 + \varepsilon/2 = \varepsilon.$$

So, choosing $L$ equal to the max of $N, M$, we see that given infinite small $\varepsilon > 0$ we can always choose $L$ so that for $n > L$, $|\Psi_{1,n} - \Psi_{3,n}| < \varepsilon$. This means that $|\Psi_{1,n} - \Psi_{3,n}| \to_{\#} 0_{\mathbb{R}_c^\#}$, i.e. {$\Psi_{1,n}$}$_{n=0}^{\infty}$ is related to {$\Psi_{3,n}$}$_{n=0}^{\infty}$.
So, we really have equivalence relation, and so by Corollary 22.1 the set $\mathbb{R}_{*\infty}$. Is partitioned into disjoint subsets (equivalence classes).

**Definition 22.17** The hyperreal numbers $*\mathbb{R}^\#_c$ are the equivalence classes $[\{\Psi_n\}_{n=0}^\infty]$ of Cauchy hyper infinite sequences of, as per Definition 22.16. That is, each such equivalence class is a hyperreal number in $*\mathbb{R}^\#_c$.

**Definition 22.18** Let $\{a_n\}_{n=0}^\infty \in *\mathbb{R}^\#_c$. We define external countable sum $\tilde{Ext} - \sum_{n=0}^{n=\infty} a_n$ by:

$$ (i) \quad \tilde{Ext} - \sum_{n=0}^{n=\infty} a_n \triangleq (b_0, b_1, \dots, b_k, \dots, \{b_n\}_{n=0}^\infty, \{b_n\}_{n=0}^\infty, \dots) = \{\tilde{b}_n\}_{n=0}^\infty, \quad (22.15) $$

where $b_0 = a_0, \dots, b_k, \dots; b_k = \text{Ext}-\sum_{n=0}^{n=k} a_n, k \in \mathbb{N}$.

(ii) We set

$$ \tilde{Ext} - \sum_{n=0}^{n=\infty} a_n \equiv \text{Ext}-\sum_{n=0}^{n=\infty} a_n. \quad (22.16) $$

**Remark 22.2** Note that in non-Archimedean field $*\mathbb{R}^\#_c$ external countable sum $\text{Ext}-\sum_{n=0}^{n=\infty} a_n$ is not preserved in general case under the substitution $a_n = b_n, n \in \mathbb{N}$, i.e. $a_n = b_n, n \in \mathbb{N} \Rightarrow_{s,w} \text{Ext}-\sum_{n=0}^{n=\infty} a_n = \text{Ext}-\sum_{n=0}^{n=\infty} b_n$. For example in non-Archimedean field $*\mathbb{R}^\#_c$ for countable sequence: $\alpha, \alpha r, \alpha r^2, \dots, \alpha r^n, \dots, n \in \mathbb{N}, r < 1$ we get

$$ \text{Ext}-\sum_{n=0}^{n=\infty} \alpha r^n = \tilde{Ext} - \sum_{n=0}^{n=\infty} \alpha r^n = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}} (1_{1_*\mathbb{R}^\#_c-r} - \overline{\{r^n\}_{n=1}^{n=\infty}}). $$

However in non-Archimedean field $*\mathbb{R}^\#_c$ one obtains that

$$ \text{Ext}-\sum_{n=0}^{n=\infty} \alpha r^n = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}}, $$

see Example 22.1.

**Example 22.1** Consider countable sequence: $\alpha, \alpha r, \alpha r^2, \dots, \alpha r^n, \dots, n \in \mathbb{N}, r < 1$.

$$ \tilde{Ext} - \sum_{n=0}^{n=\infty} \alpha r^n = \alpha \left\{ \frac{1_{1_*\mathbb{R}^\#_c-r^n}}{1_{1_*\mathbb{R}^\#_c-r}} \right\}_{n=0}^{n=\infty} = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}} - \alpha \left\{ \frac{\overline{r^n}}{1_{1_*\mathbb{R}^\#_c-r}} \right\}_{n=1}^{n=\infty} = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}} - \alpha \frac{\overline{\{r^n\}_{n=1}^{n=\infty}}}{1_{1_*\mathbb{R}^\#_c-r}}. \quad (22.17) $$

Notice that $\overline{\{r^n\}_{n=1}^{n=\infty}} \in \Delta_\omega^{+\perp 0}$.

**Definition 22.19** Let $\{a_n\}_{n=0}^{n=m} \in GF_{*\infty}, m \in *\mathbb{N}\backslash\mathbb{N}$. We define external hyper finite sum $\tilde{Ext} - \sum_{n=0}^{n=m} a_n$ by:

$$ (i) \quad \tilde{Ext} - \sum_{n=0}^{n=m} a_n \triangleq (b_0, b_1, \dots, b_k, \dots, \{b_n\}_{n=0}^{n=m}, \{b_n\}_{n=0}^{n=m}, \dots) = \{\tilde{b}_n\}_{n=0}^{n=m}, \quad (22.18) $$

where $b_0 = a_0, \dots, b_k, \dots, b_m; b_k = \text{Ext}-\sum_{n=0}^{n=k} a_n, k \in *\mathbb{N}$.

(ii) We set

$$ \tilde{Ext}-\sum_{n=0}^{n=m} a_n \equiv \tilde{Ext}-\sum_{n=0}^{n=m} a_n, \quad (22.19) $$

where $\tilde{Ext}-\sum_{n=0}^{n=m} a_n \triangleq \tilde{c}_m, c_m = \text{Ext} - \sum_{n=0}^{n=m} a_n$.

**Example 22.2** Consider hyper finite sequence: $\alpha, \alpha r, \alpha r^2, \dots, \alpha r^m, m \in *\mathbb{N}\backslash\mathbb{N}, r < 1$.

$$ \tilde{Ext} - \sum_{n=0}^{n=m} \alpha r^n = \alpha \left\{ \frac{1_{1_*\mathbb{R}^\#_c-r^n}}{1_{1_*\mathbb{R}^\#_c-r}} \right\}_{n=0}^{n=m} = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}} - \alpha \left\{ \frac{\overline{r^n}}{1_{1_*\mathbb{R}^\#_c-r}} \right\}_{n=1}^{n=m} = \alpha \frac{1_{1_*\mathbb{R}^\#_c-r}}{1_{1_*\mathbb{R}^\#_c-r}} - \alpha \frac{\overline{\{r^n\}_{n=1}^{n=m}}}{1_{1_*\mathbb{R}^\#_c-r}}. \quad (22.20) $$

From (22.17) and (22.20) we obtain

$$ \tilde{Ext} - \sum_{n \in *\mathbb{N}\backslash\mathbb{N}, n \le m} \alpha r^n = \frac{\alpha}{1_{1_*\mathbb{R}^\#_c-r}} (\overline{\{r^n\}_{n=1}^{n=\infty}} - r^m) > 0_{1_*\mathbb{R}^\#_c}. \quad (22.21) $$
**Example 22.3** Consider hyper infinite sequence: , $\alpha r, \alpha r^2, \dots, \alpha r^n, \dots, n \in \mathbb{N}, r < 1$.
$n \in *\mathbb{N}$. Type equation here.

$$
\begin{align}
\widetilde{E}xt - \sum_{n \in *\mathbb{N}} \alpha r^n &\triangleq \#\text{-} \lim_{m \to *\infty} (\widetilde{E}xt - \sum_{n=0}^{n=m} \alpha r^n) = \alpha \frac{1_{*\widetilde{\mathbb{R}}_c^\#}}{1_{*\widetilde{\mathbb{R}}_c^\#} - r} - \left( \#\text{-} \lim_{m \to *\infty} \alpha \left\{ \frac{\widetilde{r^n}}{1_{*\widetilde{\mathbb{R}}_c^\#} - r} \right\}_{n=1}^{n=m} \right) = \nonumber \\
&= \alpha \frac{1_{*\widetilde{\mathbb{R}}_c^\#}}{1_{*\widetilde{\mathbb{R}}_c^\#} - r}. \tag{22.22}
\end{align}
$$

From (22.17) and (22.22) we obtain

$$
\widetilde{Ext} - \sum_{n \in *\mathbb{N}} \alpha r^n = \overline{\{r^n\}}_{n=1}^{n=\infty} > 0_{*\overline{\mathbb{R}}_c^\#}. \quad (22.23)
$$

**Definition 22.20** Let $s, t \in *\mathbb{R}_c^\#$, so there are Cauchy hyper infinite sequences $\{\Psi_n\}_{n=0}^\infty$ and $\{\Phi_n\}_{n=0}^\infty$ with
$s = \{\Psi_n\}_{n=0}^\infty$ and $t = \{\Phi_n\}_{n=0}^\infty$.

(i) Define $s+t$ to be the equivalence class of the hyper infinite sequence $\{\Psi_n + \Phi_n\}_{n=0}^\infty$.

(ii) Define $s \times t$ to be the equivalence class of the hyper infinite sequence $\{\Psi_n \times \Phi_n\}_{n=0}^\infty$.

**Proposition 22.5** The operations $+$, $\times$ in Definition 22.18 (i),(ii) are well-defined.

Proof. (i) Suppose that $\{\Psi_n\}_{n=0}^\infty = \{\Psi_{1,n}\}_{n=0}^\infty$ and $\{\Phi_n\}_{n=0}^\infty = \{\Phi_{1,n}\}_{n=0}^\infty$. Thus means that $\Psi_n - \Psi_{1,n} \to_* 0_{*\mathbb{R}_c^\#}$, and $\Phi_n - \Phi_{1,n} \to_* 0_{*\mathbb{R}_c^\#}$ as $n \to *\infty$. Then $(\Psi_n + \Phi_n) - (\Psi_{1,n} + \Phi_{1,n}) = (\Psi_n - \Psi_{1,n}) + (\Phi_n - \Phi_{1,n})$. Now, using the familiar $\frac{\epsilon}{2}$ trick, you can construct a proof that this $\#$-tends to $0_{*\mathbb{R}_c^\#}$, and so $[\{\Psi_n + \Phi_n\}_{n=0}^\infty] = [\{\Psi_{1,n} + \Phi_{1,n}\}_{n=0}^\infty]$.

(ii) Again, suppose that $\{\Psi_n\}_{n=0}^\infty = \{\Psi_{1,n}\}_{n=0}^\infty$ and $\{\Phi_n\}_{n=0}^\infty = \{\Phi_{1,n}\}_{n=0}^\infty$. We wish to show that $[\{\Psi_n \times \Phi_n\}_{n=0}^\infty] = [\{\Psi_{1,n} \times \Phi_{1,n}\}_{n=0}^\infty]$, or, in other words, that $\Psi_n \times \Phi_n - \Psi_{1,n} \times \Phi_{1,n} \to_* 0_{*\mathbb{R}_c^\#}$ as $n \to *\infty$. Well, we add and subtract one of the other cross terms, say $\Phi_n \times \Psi_{1,n}$:

$$
\Psi_n \times \Phi_n - \Psi_{1,n} \times \Phi_{1,n} = \Psi_n \times \Phi_n + (\Phi_n \times \Psi_{1,n} - \Phi_n \times \Psi_{1,n}) - \Psi_{1,n} \times \Phi_{1,n} = (\Psi_n \times \Phi_n - \Phi_n \times \Psi_{1,n}) + \\
+ (\Phi_n \times \Psi_n - \Phi_n \times \Psi_{1,n}) + (\Phi_n \times \Psi_{1,n} - \Psi_{1,n} \times \Phi_{1,n}) = \Phi_n \times (\Psi_n - \Psi_{1,n}) + \Psi_{1,n} \times (\Phi_n - \Phi_{1,n}).
$$

Hence, we have $|\Psi_n \times \Phi_n - \Psi_{1,n} \times \Phi_{1,n}| \le |\Phi_n| \times |\Psi_n - \Psi_{1,n}| + |\Psi_{1,n}| \times |\Phi_n - \Phi_{1,n}| \le R(|\Psi_n - \Psi_{1,n}| + |\Phi_n - \Phi_{1,n}|)$. Now, noting that both $\Psi_n - \Psi_{1,n}$ and $\Phi_n - \Phi_{1,n}$ $\#$-tend to $0_{*\mathbb{R}_c^\#}$ as $n \to *\infty$ and using the $\epsilon/2$ trick (actually, this time we'll want to use $\epsilon/2R$, we see that $\Psi_n \times \Phi_n - \Psi_{1,n} \times \Phi_{1,n} \to_* 0_{*\mathbb{R}_c^\#}$ as $n \to *\infty$).

**Theorem 22.3** (i) Given any hyperreal number $s \in *\mathbb{R}_c^\#$, $s = \{\Psi_n\}_{n=0}^\infty$ such that $s \neq [0_{*\mathbb{R}_c^\#}]$ and there is $N$ such that for all $n > N$, $\Psi_n$ is multiplicative invertible, then there is a hyperreal number $t \in *\mathbb{R}_c^\#$ such that $s \times t = 1_{*\mathbb{R}_c^\#} \triangleq 1_{*\widetilde{\mathbb{R}}_c^\#}$. (ii) Given any hyperreal number $s \in *\mathbb{R}_c^\#$, $s = \{\Psi_n\}_{n=0}^\infty$ such that $s \neq [0_{*\mathbb{R}_c^\#}]$ and there is $N$ such that for all $n > N$, $\Psi_n$ is multiplicative semi-invertible, then there is a hyperreal number $t \in *\mathbb{R}_c^\#$ such that $s \times t = [\omega_n]$, where $\omega_n \equiv 1_{*\mathbb{R}_c^\#}$ if $n = n_i$ and $\omega_n \equiv 0_{*\mathbb{R}_c^\#}$ if $n \neq n_i, i \in *\mathbb{N}$.

Proof. (i) First we must properly understand what the theorem says. The premise is that $s$ is nonzero, which means that $s$ is not in the equivalence class of $0_{*\overline{\mathbb{R}}_c^\#} \triangleq 0_{*\widetilde{\mathbb{R}}_c^\#}$. In other words, $s = \{\Psi_n\}_{n=0}^\infty$ where $\{\Psi_n\}_{n=0}^\infty - 0_{*\widetilde{\mathbb{R}}_c^\#}$ does not $\#$-converge to $0_{*\overline{\mathbb{R}}_c^\#}$ as $n \to *\infty$. From this, we are to deduce the existence of a hyperreal number $t = \{\Phi_n\}_{n=0}^\infty$ such that $s \times t = [\{\Psi_n \times \Phi_n\}_{n=0}^\infty]$ is the same equivalence class as $1_{*\widetilde{\mathbb{R}}_c^\#}$. Doing so is actually an easy consequence of the fact that nonzero hyperreal numbers from $*\overline{\mathbb{R}}_c^\#$ have multiplicative inverses, but there is a subtle difficulty. Just because $s$ is nonzero (i.e. $\{\Psi_n\}_{n=0}^\infty$ does not $\#$-tend to $0_{*\overline{\mathbb{R}}_c^\#}$ as $n \to *\infty$, there's no reason any number of the terms in $\{\Psi_n\}_{n=0}^\infty$ can't equal $0_{*\overline{\mathbb{R}}_c^\#}$. However, it turns out that eventually, $\{\Psi_n\}_{n=0}^\infty \neq 0_{*\overline{\mathbb{R}}_c^\#}$ by
Lemma 22.1.

Proof. (ii) Immediately from definitions and by Lemma 22.1.

**Lemma 22.1** If {$\Psi_n$}$_{n=0}^{*\infty}$ is a Cauchy hyper infinite sequence which does not #-tend to $0_{*\mathbb{R}^\#}$, then there is an $N \in *\mathbb{N}$ such that: (i) for all $n > N$, $\Psi_n \neq 0_{*\mathbb{R}^\#}$ or (ii) there is hyper infinite subsequence {$\Psi_{n_i}$}$_{n_i=0}^{*\infty} \subset \{\Psi_n\}^{*\infty}_{n=0}$ such that {$\Psi_{n_i}$}$_{n_i=M}^{*\infty} \neq 0_{*\mathbb{R}^\#}$, $i \in *\mathbb{N}$, $M \in *\mathbb{N} \setminus \mathbb{N}$.

We will now use it to complete the proof of Theorem 22.3.

Let $N \in *\mathbb{N}$ be such that $\Psi_n \neq 0_{*\mathbb{R}^\#}$ for $n > N$. Define hyper infinite sequence {$\Phi_n$}$_{n=0}^{*\infty}$ of hyperreal numbers from $*\mathbb{R}^\#_c$ as follows: for $n \le N$, $\Phi_n = 0_{*\mathbb{R}^\#_c}$, and for $n > N$, $\Phi_n = \frac{1_{*\mathbb{R}^\#_c}}{\Psi_n}$. Thus we obtain hyper infinite sequence {$\Phi_n$}$_{n=0}^{*\infty} = (0_{*\mathbb{R}^\#_c}, 0_{*\mathbb{R}^\#_c}, \dots, 0_{*\mathbb{R}^\#_c}, \frac{1_{*\mathbb{R}^\#_c}}{\Psi_{N+1}}, \frac{1_{*\mathbb{R}^\#_c}}{\Psi_{N+2}}, \dots)$. This makes sense since, for $n > N$, an is a nonzero hyperreal number, so $1_{*\mathbb{R}^\#_c}/\Psi_n$ exists. Then $\Psi_n \times \Phi_n$ is equal to $\Psi_n \times 0_{*\mathbb{R}^\#_c} = 0_{*\mathbb{R}^\#_c}$ for $n \le N$, and equals $\Psi_n \times \Phi_n = \Psi_n \times 1_{*\mathbb{R}^\#_c}/\Psi_n = 1_{*\mathbb{R}^\#_c}$ for $n > N$. Well, then, if we look at the hyper infinite sequence $1_{*\mathbb{R}^\#_c} \triangleq 1_{*\mathbb{R}^\#_c}$, we have $1_{*\mathbb{R}^\#_c} - \{\Psi_n \times \Phi_n\}^{*\infty}_{n=0}$ is the hyper infinite sequence which is $1_{*\mathbb{R}^\#_c} - 0_{*\mathbb{R}^\#_c} = 1_{*\mathbb{R}^\#_c}$ for $n \le N$ and equals $1_{*\mathbb{R}^\#_c} - 1_{*\mathbb{R}^\#_c} = 0_{*\mathbb{R}^\#_c}$ for $n > N$. Since this hyper infinite sequence is eventually equal to $0_{*\mathbb{R}^\#_c}$, it #-converges to $0_{*\mathbb{R}^\#_c}$ as $n \to *\infty$, and so $[\{\Psi_n \times \Phi_n\}^{*\infty}_{n=0}] = 1_{*\mathbb{R}^\#_c}$. This shows that $t = [\{\Phi_n\}^{*\infty}_{n=0}]$ is a multiplicative inverse to $s = [\{\Psi_n\}^{*\infty}_{n=0}]$.

**Definition 22.21** Let $s \in *\mathbb{R}^\#_c$. Say that $s$ is positive if $s \neq 0_{*\mathbb{R}^\#_c}$, and if $s = [\{\Psi_n\}^{*\infty}_{n=0}]$ for some Cauchy hyper infinite sequence such that for some $N$, $\Psi_n > 0_{*\mathbb{R}^\#_c}$ for all $n > N$. Given two hyperreal numbers $s, t \in *\mathbb{R}^\#_c$, say that $s > t$ if $s - t$ is positive.

**Theorem 22.3** Let $s, t \in *\mathbb{R}^\#_c$ be hyperreal numbers such that $s > t$, and let $r \in *\mathbb{R}^\#_c$. Then $s + r > t + r$.

Proof. Let $s = [\{\Psi_n\}^{*\infty}_{n=0}]$, $t = [\{\Phi_n\}^{*\infty}_{n=0}]$ and $r = [\{\Theta_n\}^{*\infty}_{n=0}]$. Since $s > t$, i.e. $-t > 0_{*\mathbb{R}^\#_c}$, we know that there is an $N \in *\mathbb{N}$ such that, for $n > N$, $\Psi_n - \Phi_n > 0_{*\mathbb{R}^\#_c}$. So $\Psi_n > \Phi_n$ for $n > N$. Now, adding $\Theta_n$ to both sides of this inequality, we have $\Psi_n + \Theta_n > \Phi_n + \Theta_n$ for $n > N$, or $(\Psi_n + \Theta_n) - (\Phi_n + \Theta_n) > 0_{*\mathbb{R}^\#_c}$ for $n > N$. Note also that $(\Psi_n + \Theta_n) - (\Phi_n + \Theta_n) = \Psi_n - \Phi_n$ does not #-converge to $0_{*\mathbb{R}^\#_c}$ as $n \to *\infty$, by the assumption that $s - t > 0_{*\mathbb{R}^\#_c}$. Thus, by Definition 22.19, this means that: $s + r = [\{\Psi_n + \Theta_n\}^{*\infty}_{n=0}] > [\{\Psi_n + \Phi_n\}^{*\infty}_{n=0}] = t + r$.

**Remark 22.3** There is canonical imbedding

$$ *\mathbb{R}^\#_c \hookrightarrow *\mathbb{R}^\#_c \qquad (22.24) $$

defined by $i: a \mapsto [\tilde{a}]$.

## 23. Generalized convergence of hyper infinite sequences of #-closed operators

When we consider various perturbation problems related to #-closed operators, it is necessary to make precise what is meant by a "small" perturbation. This can be done in a most natural way by introducing a metric in the set $W(X, Y)$ of all #-closed operators from $X$ to $Y$. If $T, S \in W(X, Y)$, their graphs $G(T), G(S)$ are #-closed linear manifolds in the product space $X \times Y$. Thus the "distance" between $T$ and $S$ can be measured by the "gap" between the closed linear manifolds. $G(T), G(S)$. In this way we are led to consider how to measure the gap of two #-closed linear manifolds of a non-Archimedean Banach space. In this paragraph we shall consider closed linear manifolds $M, N, \dots$ of a non-Archimedean Banach space $Z$. We denote by $S_M$ the unit sphere of $M$ (the set of all $u \in M$ with $\|u\|_# = 1$). For any two #-closed linear manifolds $M, N$ of $Z$, we set

$$ \delta(M, N) = \sup_{u \in S_M} \mathrm{dist}(u, N), \qquad (23.1) $$

$$ \delta(M, N) = \max[\delta(M, N), \delta(N, M)]. \qquad (23.2) $$
Note that (23.1) has no meaning if $M = \emptyset$; in this case we define $\delta(0, N) = 0_{*\mathbb{R}_c^\#}$ for any $N$. On the other hand $\delta(M, 0) = 1_{*\mathbb{R}_c^\#}$ if $M \neq \emptyset$, as is seen from the definition. $\delta(M, N)$ can also be characterized as the smallest number $\delta$ such that

$$ \mathrm{dist}(u, N) \leq \delta \|u\|_{\#} \qquad (23.3) $$

for all $u \in M$.

**Definition 23.1** The quantity $\hat{\delta}(M, N)$ is called the gap between $M, N$. The following relations follow directly from the definition.

$$ \delta(M, N) = 0_{*\mathbb{R}_c^\#} \text{ if and only if } M \subset N. \qquad (23.4) $$

$$ \hat{\delta}(M, N) = 0_{*\mathbb{R}_c^\#} \text{ if and only if } M = N. \qquad (23.5) $$

$$ \hat{\delta}(M, N) = \hat{\delta}(N, M). \qquad (23.6) $$

$$ 0_{*\mathbb{R}_c^\#} \leq \delta(M, N) \leq 1_{*\mathbb{R}_c^\#}, \quad 0_{*\mathbb{R}_c^\#} \leq \hat{\delta}(M, N) \leq 1_{*\mathbb{R}_c^\#}. \qquad (23.7) $$

(23.5) and (23.6) suggest that $\delta(M, N)$ could be used to define a distance between $M$ and $N$. But this is not possible, since the function $\delta(X, Y)$ does not in general satisfy the triangle inequality required of a distance function. This inconvenience may be removed by modifying the definition (23.1)-(23.2). We set

$$ d(M, N) = \sup_{u \in S_M} \mathrm{dist}(u, S_N), \qquad (23.8) $$

$$ \hat{d}(M, N) = \max[d(M, N), d(N, M)]. \qquad (23.9) $$

(23.8) does not make sense if either $M$ or $N$ is 0. In such cases we set

$$ d(0, N) = 0_{*\mathbb{R}_c^\#} \text{ for any } N; d(M, 0) = 2_{*\mathbb{R}_c^\#} \text{ for } M \neq \emptyset. \qquad (23.10) $$

Furthermore, $d$ and $\hat{d}$ satisfy the triangle inequalities:

$$ d(L, N) \leq d(L, M) + d(M, N), \quad \hat{d}(L, N) \leq \hat{d}(L, M) + \hat{d}(M, N). \qquad (23.11) $$

The second inequality of (23.11) follows from the first, which in turn follows easily from the definition. The proof will be left to the reader. The case when some of $L, M, N$ are 0 should be considered separately; note (23.10). The set of all #-closed linear manifolds of $Z$ becomes a metric space endowed with $*\mathbb{R}_c^\#$-valued metric if the distance between $M, N$ is defined by $\hat{d}(L, N)$.

**Definition 23.2** A hyper infinite sequence $M_n, n \in *\mathbb{N}$ of #-closed linear manifolds #-converges to $M$ if $\hat{d}(M_n, M) \to_* 0_{*\mathbb{R}_c^\#}$ as $n \to *\infty$. Then we write $M_n \to_* M$ or $\#\mathrm{lim}_{n \to *\infty} M_n = M$.

**Remark 23.1** Although the gap $\hat{\delta}$ is not a proper distance function, it is more convenient than the proper distance function $\hat{d}$ for applications since its definition is slightly simpler. Furthermore, when we consider the topology of the set of all closed linear manifolds, the two functions give the same result. This is due to the following inequalities

$$ \delta(M, N) \leq d(M, N) \leq 2\delta(M, N), \qquad (23.12) $$

$$ \hat{\delta}(M, N) \leq \hat{d}(M, N) \leq 2\hat{\delta}(M, N). \qquad (23.13) $$

We set

$$ \delta(T, S) = \delta(G(T), G(S)), \quad \hat{\delta}(T, S) = \hat{\delta}(G(T), G(S)) = \max[\delta(T, S), \delta(S, T)]. \qquad (23.14) $$
The quantity $\hat{\delta}(T, S)$ will be called the gap between $T$ and $S$. Similarly we can define the distance $\hat{d}(T, S)$ between $T$ and $S$ as equal to $\hat{d}(G(T), G(S))$. Under this distance function $W(X, Y)$ becomes a non-Archimedean metric space endowed with $\mathbb{R}_c^\#$-valued metric. In this space the $\#$-convergence of a hyper infinite sequence $T_n \in W(X, Y)$ to a $T \in W(X, Y)$ as $n \to \infty$ is defined by $\hat{d}(T_n, T) \to_\# 0_{\mathbb{R}_c^\#}$. But since $\hat{\delta}(T, S) \le \hat{d}(T, S) \le 2\hat{\delta}(T, S)$ in virtue of (23.13), this is true if and only if $\hat{\delta}(T_n, T) \to_\# 0_{\mathbb{R}_c^\#}$ as $n \to \infty$.

**Definition 23.3** Let $T_n \in W(X, Y)$, $n \in \mathbb{N}$. If $\hat{\delta}(T_n, T) \to_\# 0_{\mathbb{R}_c^\#}$ as $n \to \infty$ we shall say that the operator $T_n$ $\#$-converges to $T$ or $T_n \to_\# T$ in the generalized sense.

**Theorem 23.1** Let $T: H^\# \to H^\#$ be a $\#$-selfadjoint operator. Then there is a $\delta > 0$ such that any $\#$-closed symmetric operator $S$ with $\hat{\delta}(S, T) < \delta$ is $\#$-selfadjoint, where $\hat{\delta}(S, T)$ denotes the gap between $S$ and $T$.

**Corollary 23.1** Let $T, T_n$. be $\#$-closed symmetric operators and let hyper infinite $T_n, n \in \mathbb{N}$. $\#$-converge to $T$ in the generalized sense. If $T$ is $\#$-selfadjoint, then $T_n$ is $\#$-selfadjoint for sufficiently large $n \in \mathbb{N}$.

## 24. Strong convergence of the resolvent

Let $T_n, n \in \mathbb{N}, T_n \in W(X, X)$ be a hyper infinite sequence of $\#$-closed operators in a non-Archimedean Banach space $X$. In this section we are briefly concerned with general considerations on strong $\#$-convergence of the resolvents $R_n(\xi) = (T_n - \xi)^{-1}_{\mathbb{R}_c^\#}$. We remind the fundamental result on the $\#$-convergence in $\#$-norm of the resolvents: if $R_n(\xi)$ $\#$-converges in $\#$-norm to the resolvent $R(\xi) = (T - \xi)^{-1}_{\mathbb{R}_c^\#}$ of a $\#$-closed operator $T$ for some $\xi \in P(T)$, then the same is true for every $\xi \in P(T)$ (see Theorem 23.2 and Remark 23.2). There is no corresponding theorem for strong $\#$-convergence of the resolvents. Nevertheless, we can prove several theorems on the set of points $\xi$ where the $R_n(\xi)$ are strongly $\#$-convergent or bounded. Let us define the region of boundedness, denoted by $\Delta_b$, for the hyper infinite sequence $R_n(\xi)$ as the set of all complex numbers $\xi \in \mathbb{C}_c^\#$ such that $\xi \in P(T_n)$ for sufficiently large $n \in \mathbb{N}$ and the sequence $\| R_n(\xi) \|_\#$ is bounded [for $n$ so infinite large that the $R_n(\xi)$ are defined]. Furthermore, let $\Delta_s$ be the set of all $\xi$ such that $s$-$\#$-lim$_{n \to \infty} R_n(\xi) = R'(\xi)$ exists. $\Delta_s$ will be called the region of strong $\#$-convergence for $R_n(\xi)$. Similarly we define the region $\Delta_b$ of $\#$-convergence in $\#$-norm for $R_n(\xi)$. Obviously we have $\Delta_u \subset \Delta_s \subset \Delta_b$.

**Theorem 24.1** $\Delta_b$ is an $\#$-open set in the complex plane $\mathbb{C}_c^\#$. Operator $R_n(\xi)$ is bounded in $\mathbb{R}_c^\#$ uniformly in $n$ and $\xi$ in any $\#$-compact subset of $\Delta_b$.

**Proof.** Let $\xi_0 \in \Delta_b$; for $|\xi - \xi_0| < 1_{\mathbb{R}_c^\#}/\| R_n(\xi_0) \|_\#$ = $(\| R_n(\xi_0) \|_\#)^{-1}_{\mathbb{R}_c^\#}$ we have the Neumann hyper infinite series

$$R_n(\xi) = \mathrm{Ext} - \sum_{k=0}^{\infty} (\xi - \xi_0) [\| R_n(\xi_0) \|_\#]^k. \quad (24.1)$$

If $\| R_n(\xi_0) \|_\# \le M_0$, then $\| R_n(\xi_0) \|_\# \le M_0(1_{\mathbb{R}_c^\#} - M_0 |\xi - \xi_0|^{-1}_{\mathbb{R}_c^\#})$ for $|\xi - \xi_0| < M_0^{-1}_{\mathbb{R}_c^\#}$. The theorem follows immediately.

**Remark 24.1** Theorem 24.1 implies that $\Delta_b$ consists of at most a hyper infinite number of $\#$-connected $\#$-open sets $\Delta_{b_1}, \Delta_{b_2}, \dots$, (the components of $\Delta_b$).

**Theorem 24.2** $\Delta_s$ is relatively $\#$-open and $\#$-closed in $\Delta_b$ (so that $\Delta_s$ is the union of some of the components $\Delta_{b_k}$ of $\Delta_b$). The strong $\#$-convergence $R_n(\xi) \to_\# R(\xi)$ is uniform in each $\#$-compact subset of $\Delta_s$.

**Remark 24.2** For convenience we call $\Delta_b$ also the region of boundedness for the sequence if there is no possibility of confusion. Similarly for $\Delta_s$ and $\Delta_b$.

**Remark 24.3** The strong $\#$-convergence $R_n(\xi) \to_\# R(\xi)$ is uniform in $\xi$ if $\| R_n(\xi) - R(\xi) \|_\# \to_\# 0_{\mathbb{R}_c^\#}$ uniformly in $\xi$ for each fixed $u \in X$.

**Remark 24.4** The strong $\#$-limit $R'(\xi)$ of $R_n(\xi)$ for $\xi \in \Delta_s$ need not be the resolvent of an operator. In any case, however, $R'(\xi)$ satisfies the resolvent equation

$$R'(\xi_1) - R'(\xi_2) = (\xi_1 - \xi_2)R'(\xi_1)R'(\xi_2), \xi_1, \xi_2 \in \Delta_s, \quad (24.2)$$

as the strong $\#$-limit of operators $R_n(\xi)$ which satisfy the same equation.

**Definition 24.1** For this reason $R'(\xi)$ is called a pseudo-resolvent. Note that $R'(\xi_1)$ and $R'(\xi_2)$ commute.
**Remark 24.5** The Eq.(24.2) implies that the null space $N = N(R'(\xi))$ and the range $R = R(R'(\xi))$ of $R'(\xi)$ are independent of $\xi$. In fact, it follows from Eq.(24.2) that $R'(\xi_2)u = 0$ implies $R'(\xi_1)u = 0$ and that $u = R'(\xi_1)v$ implies $u = R'(\xi)w$ with $w = v - (\xi_1 - \xi_2)u$.

**Remark 24.6** The pseudo-resolvent $R'(\xi)$ is a resolvent of an #-closed operator $T$ if and only if $N = 0$.

**Theorem 24.3** Let $\Delta_s$ be nonempty. There are the alternatives: either $R'(\xi)$ is invertible for no $\xi \in \Delta_s$ or $R'(\xi)$ is equal to the resolvent $R(\xi) = (T - \xi)^{-1}{}^{\ast_{\mathbb{R}_c^\#}}$ of a unique operator $T \in W(X, X)$. In the latter case we have $\Delta_s = P(T) \cap \Delta_b$.

Proof. Only the last statement remains to be proved. We have $\Delta_s \subset P(T) \cap \Delta_b$ since $\Delta_s \subset P(T)$. To prove opposite inclusion, we note the identity

$$R_n(\xi) - R(\xi) = (1_{\mathbb{R}_c^\#} + (\xi - \xi_0) R_n(\xi))(R_n(\xi_0) - R(\xi_0))(1_{\mathbb{R}_c^\#} + (\xi - \xi_0)R(\xi)) \quad (24.3)$$

for $\xi, \xi_0 \in P(T) \cap \Delta_b$ this is a simple consequence of the resolvent equations for $R_n(\xi)$ and $R(\xi)$. If $\xi_0 \in \Delta_s$, we have s-#-lim $R_n(\xi_0) = R'(\xi_0) = R(\xi_0)$ so that Eq.(24.3) gives s-#-lim $R_n(\xi) = R(\xi)$ by the uniform boundedness of hyper infinite sequence $R_n(\xi)$, $n \in \mathbb{N}$. This shows that $\xi \in \Delta_s$ and completes the proof.

**Corollary 24.1** Let $T_n, n \in \mathbb{N}$ and $T$ be sebe #-selfadjoint operators in a non-Archimedean Hilbert space, with the resolvents $R_n(\xi)$ and $R(\xi)$. If s-#-lim$_{n \to \infty} R_n(\xi) = R(\xi)$ for some complex number $\xi \in \mathbb{C}_c^\#$, then the same is true for every nonreal $\xi \in \mathbb{C}_c^\#$.

Proof. Since $\| R_n(\xi) \|_\# \le 1_{\mathbb{R}_c^\#} / |\mathrm{Im}\xi|$, all nonreal numbers $\xi$ are included in $\Delta_b$ as well as in $P(T)$. Thus $P(T) \cap \Delta_b$ also includes all nonreal $\xi$, and the assertion follows from Theorem 24.3.

**Definition 24.2** When the second alternative of Theorem 24.3 is realized, we shall say that $R_n(\xi)$ #-converges strongly to $R(\xi)$ on $\Delta_s$, and that $T_n$ #-converges strongly to $T$ in the generalized sense.

A criterion for generalized strong convergence is given by:

**Theorem 24.5** Let $T_n, T \in W(X, X)$ and let there be a #-core $D$ of $T$ such that each $u \in D$ belongs to $D(T_n)$ for sufficiently infinite large $n$ and $T_n u \to_\# Tu$. If $P(T) \cap \Delta_b$ is not empty, $T_n$ #-converges strongly to $T$ in the generalized sense and $\Delta_s = P(T) \cap \Delta_b$.

## 25. Conclusion

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator $\varphi(x, t)$ no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian $P(\varphi)_4$ exists and that the canonical $C^*$-algebra of bounded observables corresponding to this model satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the $\lambda(\varphi^4)_4$ quantum field theory model is Lorentz covariant. For each Poincare transformation $a, \Lambda$ and each bounded region $O$ of Minkowski space we obtain a unitary operator $U$ which correctly transforms the field bilinear forms $\varphi(x, t)$ for $(x, t) \in O$. The von Neumann algebra $\mathfrak{C}(O)$ of local observables is obtained as standard part of external nonstandard algebra $\mathcal{B}_\#(O)$.

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