nyuuzyou/wikis huggingface dataset:
Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M2

== Theorem ==
Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A \times A \to \R$ be the real-valued function defined as:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$
Then $d_2$ satisfies .
== Proof ==
Let $H = \struct{\ell^2, d_{\ell^2}}$ denote the Hilbert sequence space, where:
:$\ell^2$ denotes the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent
:$d_{\ell^2}$ denotes the real $2$-sequence metric, that is, the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
::$\ds \forall x = \sequence {x_n}, y = \sequence {y_n} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{n \mathop = 0}^\infty \paren {x_n - y_n}^2}^{\frac 1 2}$
=== Lemma ===
Let $x_1, x_2, x_3 \in A$.
From Lemma:
:$\exists y_1, y_2, y_3 \in \ell^2 : $
::$\forall i, j \in \set{1, 2, 3} : \map {d_2} {x_i, x_j} = \map {d_{\ell^2}} {y_i, y_j}$
We have:
The result follows.
Category:Generalized Hilbert Sequence Space is Metric Space
Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M3

== Theorem ==
Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A \times A \to \R$ be the real-valued function defined as:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$
Then $d_2$ satisfies .
== Proof ==
Let $H = \struct{\ell^2, d_{\ell^2}}$ denote the Hilbert sequence space, where:
:$\ell^2$ denotes the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent
:$d_{\ell^2}$ denotes the real $2$-sequence metric, that is, the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
::$\ds \forall x = \sequence {x_n}, y = \sequence {y_n} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{n \mathop = 0}^\infty \paren {x_n - y_n}^2}^{\frac 1 2}$
=== Lemma ===
Let $x_1, x_2 \in A$.
From Lemma:
:$\exists y_1, y_2 \in \ell^2 : $
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$ and $\map {d_2} {x_2, x_1} = \map {d_{\ell^2}} {y_2, y_1}$
We have:
The result follows.
Category:Generalized Hilbert Sequence Space is Metric Space
Generalized Hilbert Sequence Space is Metric Space/Metric Space Axiom M4

== Theorem ==
Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A \times A \to \R$ be the real-valued function defined as:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$
Then $d_2$ satisfies .
== Proof ==
Let $H = \struct{\ell^2, d_{\ell^2}}$ denote the Hilbert sequence space, where:
:$\ell^2$ denotes the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent
:$d_{\ell^2}$ denotes the real $2$-sequence metric, that is, the real-valued function $d_{\ell^2}: \ell^2 \times \ell^2: \to \R$ defined as:
::$\ds \forall x = \sequence {x_n}, y = \sequence {y_n} \in \ell^2: \map {d_{\ell^2}} {x, y} := \paren {\sum_{n \mathop = 0}^\infty \paren {x_n - y_n}^2}^{\frac 1 2}$
=== Lemma ===
Let $x_1, x_2 \in A : x_1 \ne x_2$.
From Lemma:
:$\exists y_1, y_2 \in \ell^2 :$
::$y_1 \ne y_2$
::$\map {d_2} {x_1, x_2} = \map {d_{\ell^2}} {y_1, y_2}$
We have:
The result follows.
Category:Generalized Hilbert Sequence Space is Metric Space
Generalized Hilbert Sequence Space is Metric Space/Well-Defined

== Theorem ==
Let $\alpha$ be an infinite cardinal number.
Let $I$ be an indexed set of cardinality $\alpha$.
Let $A$ be the set of all real-valued functions $x : I \to \R$ such that:
:$(1)\quad \set{i \in I: x_i \ne 0}$ is countable
:$(2)\quad$ the generalized sum $\ds \sum_{i \mathop \in I} x_i^2$ is a convergent net.
Let $d_2: A \times A \to \R$ be the real-valued function defined as:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{i \mathop \in I} \paren {x_i- y_i}^2}^{\frac 1 2}$
Then $d_2$ is well-defined.
== Proof ==
Let $\ell^2$ denote the real $2$-sequence space, that is, the set of all real sequences $\sequence {x_n}$ such that the series $\ds \sum_{n \mathop = 0}^\infty x_n^2$ is convergent
From Characterization of Hausdorff Property in terms of Nets:
:a convergent net in $\R$ has a unique limit. 
To show that $d_2$ is well-defined, it is sufficient to show:
:$\ds \forall x = \family {x_i}, y = \family {y_i} \in A:$ the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i - y_i}^2$ converges
Let $x = \family {x_i}, y = \family {y_i} \in A$.
Let $\ds \sum_{i \mathop \in I} x_i^2$ and $\ds \sum_{i \mathop \in I} y_i^2$ converge to $a, b \in \R$ respectively. 
From Characterization of Generalized Hilbert Sequence Space:
:there exists enumeration $J = \set{j_0, j_1, j_2, \ldots}$ of a countable set of $I$:
::$\set{i \in I : x_i \ne 0}, \set{i \in I : y_i \ne 0} \subseteq J$
::$\sequence{x_{j_n}}, \sequence{y_{j_n}} \in \ell^2$
::$\ds \sum_{n \mathop = 0}^\infty x_{j_n}^2 = a, \sum_{n \mathop = 0}^\infty y_{j_n}^2 = b$
From P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space:
:$\sequence{x_{j_n} - y_{j_n}} \in {\ell^2}$
We have:
:$\forall i \in I \setminus J : x_i = y_i = 0$
Hence:
:$\forall i \in I \setminus J : x_i - y_i = 0$
It follows:
:$\set{i : x_i - y_i \ne 0} \subseteq J$
From Characterization of Generalized Hilbert Sequence Space:
:$\family{x_i - y_i}_{i \mathop \in I} \in A$
By definition of $A$:
:the generalized sum $\ds \sum_{i \mathop \in I} \paren {x_i - y_i}^2$ converges
The result follows.
Category:Generalized Hilbert Sequence Space is Metric Space
Generalized Integration by Parts

== Theorem ==
Let $\map f x, \map g x$ be real functions which are integrable and at least $n$ times differentiable.
Then:
where $f^{\paren n}$ denotes the $n$th derivative of $f$.
== Proof ==
Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \int f^{\paren n} g \rd x = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^\paren j + \paren {-1}^n \int f g^{\paren n} \rd x$
=== Basis for the Induction ===
$\map P 1$ is the case:
:$\ds \int f' g \rd x = f g - \int f g' \rd x$
which is proved in Integration by Parts.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
:$\ds \int f^{\paren k} g \rd x = \sum_{j \mathop = 0}^{k - 1} \paren {-1}^j f^{\paren {k - j - 1} } g^{\paren j} + \paren {-1}^k \int f g^{\paren k} \rd x$
Then we need to show:
:$\ds \int f^{\paren {k + 1} } g \rd x = \sum_{j \mathop = 0}^k \paren {-1}^j f^{\paren {k - j} } g^{\paren j} + \paren {-1}^{k + 1} \int f g^{\paren {k + 1} } \rd x$
=== Induction Step ===
This is our induction step:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \forall n \in \N_{>0}: \int f^{\paren n} g \rd x = \sum_{j \mathop = 0}^{n - 1} \paren {-1}^j f^{\paren {n - j - 1} } g^{\paren j} + \paren {-1}^n \int f g^{\paren n} \rd x$
assuming that $f$ and $g$ are sufficiently differentiable.
== Sources ==
* : $\S 14$: General Rules of Integration: $14.48$
* : $\S 16$: Indefinite Integrals: General Rules of Integration: $16.48.$
Category:Integral Calculus
Generalized Linear Model/Examples/Bernoulli Variable

== Example of Generalized Linear Model ==
Let $Y$ be a Bernoulli variable with expectation $\mu = p$.
Then the link function $\map g \mu$ of the generalized linear model for $Y$ is given by:
:$\map g \mu = \map \ln {\dfrac p {1 - p} }$
=== Application ===
== Sources ==
* 
* 
Category:Examples of Generalized Linear Models
Generalized Linear Model/Examples/Bernoulli Variable/Application

== Example of Generalized Linear Model ==
Let $Y$ be a Bernoulli variable with expectation $\mu = p$.
Let the link function $\map g \mu$ of the generalized linear model for $Y$ be given by:
:$\map g \mu = \map \ln {\dfrac p {1 - p} }$
This can be applied to the study of the toxicity of insecticides.
Let $p_i$ be the probability that a particular insect will die when exposed to a dose $x_i$ of an insecticide.
Under a set of usual conditions:
:$\map \ln {\dfrac {p_i} {1 - p_i} } = \beta_0 + \beta_1 x$
If batches of insects are subjected to doses $x_1, x_2, \ldots, x_i, \ldots, x_n$ of the insecticide, the proportion that dies in batch $i$ provides an estimate of $p_i$.
The parameters $\beta_0$ and $\beta_1$ can be obtained by maximum likelihood estimation.
Iterative procedures are usually needed, because the variance of $\map \ln {\dfrac p {1 - p} }$ is a function of $p$.
== Sources ==
* 
* 
Category:Examples of Generalized Linear Models
Generalized Linear Model/Examples/Poisson Distribution

== Example of Generalized Linear Model ==
Let $Y$ be a random variable with a Poisson distribution.
Let $Y$ have an expectation $\lambda$ which varies with $x$.
Then the link function $\map g \lambda$ of the generalized linear model for $Y$ is given by:
:$\map g \lambda = \ln \lambda$
== Sources ==
* 
* 
Category:Examples of Generalized Linear Models
Generalized Linear Model over Normal Distribution reduces to Linear Regression Model

== Theorem ==
Let $Y$ be a random variable which obeys a normal distribution.
Let a generalized linear model for one variable for $Y$ be:
:$\mu = \expect Y = \beta_0 + \beta_1 x$
whose variance is $\sigma^2$.
Then the generalized linear model reduces to the linear regression model.
== Proof ==
== Sources ==
* 
* 
Category:Generalized Linear Models
Category:Linear Regression Model
Generalized Pentagonal Number/Sequence

== Sequence of Generalized Pentagonal Numbers ==
The sequence of generalized pentagonal numbers, for $n \in \Z_{\ge 0}$, begins:
:$GP_n = 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, \ldots$
Category:Generalized Pentagonal Numbers
Generalized Polar Decomposition in C*-Algebra

== Theorem ==
Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra. 
Let $x \in A$.
Let $\cmod x$ be the modulus of $x$. 
Let $0 < \beta < 1$ be a real number.
Then there exists $u \in A$ such that:
:$x = u \cmod x^\beta$
where $\cmod x^\beta$ is obtained from the continuous functional calculus.
== Proof ==
Let $\le_A$ be the canonical preordering of $A$.
From Product of Element of C*-Algebra with its Star is Positive, we have ${\mathbf 0}_A \le_A x^\ast x$.
Hence since $x^\ast x \le_A x^\ast x$ and $0 < \beta/2 < 1/2$, there exists $u \in A$ such that:
:$x = u \paren {x^\ast x}^{\beta/2}$
from Factorization Theorem for C*-Algebra in terms of Bound on Modulus.
From Power of Power of Positive Element of Unital C*-Algebra, we have:
:$u \paren {x^\ast x}^{\beta/2} = u \paren {\paren {x^\ast x}^{1/2} }^\beta = u \cmod x^\beta$
== Sources ==
* : Corollary $4.7$
Category:C*-Algebras
Generalized Sum Commutes with Inner Product

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be an inner product space.
Let $\Lambda$ be a set. 
Let $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a $\Lambda$-indexed family in $\HH$ such that the generalized sum $\ds \sum \set {x_\lambda : \lambda \in \Lambda}$ converges.
Let $y \in \HH$.
Then:
:$\ds \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda}$ converges 
and further:
:$\ds \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda} = \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y $ 
== Proof ==
Let $\struct {\FF, \subseteq}$ be the set of finite subsets of $\Lambda$ ordered by inclusion.
Define, for $F \in \FF$:
:$\ds \map {s_1} F = \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda}$
and:
:$\ds \map {s_2} F = \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y$
Noting that $\ds \family {\sum_{\lambda \mathop \in F} x_\lambda}_{F \in \FF}$ converges to $\ds \sum \set {x_\lambda : \lambda \in \Lambda}$, we have that:
:$\family {\map {s_2} F}_{\lambda \in \Lambda}$ converges to $\ds \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y$
by Characterization of Continuity in terms of Nets.
From the linearity of $\innerprod \cdot \cdot$ in the first argument we have:
:$\map {s_1} F = \map {s_2} F$
Hence since $\family {\map {s_2} F}_{F \in \FF}$ converges to $\ds \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y$, so does $\family {\map {s_1} F}_{F \in \FF}$.
Hence we obtain the convergence of:
:$\ds \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda}$
and obtain:
:$\ds \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda} = \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y $ 
Category:Generalized Sums
Generalized Sum Restricted to Non-zero Summands

== Theorem ==
Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$
:
:the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converges to $h$
=== Corollary ===
== Proof ==
=== Necessary Condition ===
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.
It will be shown that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$.
=== Sufficient Condition ===
Let the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converge to $h \in G$.
It will be shown that $\ds \sum_{i \mathop \in I} g_j$ converges to $h$.
Category:Generalized Sums
Category:Generalized Sum Restricted to Non-zero Summands
Generalized Sum Restricted to Non-zero Summands/Corollary

== Theorem ==
Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.
Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$
Let $h \in G$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$
:
:the generalized sum $\ds \sum_{k \mathop \in K} g_k$ converges to $h$
== Proof ==
Let $J = \set{i \in I : g_i \ne 0_G}$.
We have:
Category:Generalized Sum Restricted to Non-zero Summands
Generalized Sum Restricted to Non-zero Summands/Necessary Condition

== Theorem ==
Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g }_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.
Then:
:the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converges to $h$
== Proof ==
Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.
By definition of convergent net:
:$(1) \quad \exists F \subseteq I : F \ne \O : \forall E \subseteq I : E \supseteq F \implies \ds \sum_{i \mathop \in E} g_i \in U$ 
where $\ds \sum_{i \mathop \in E} g_i$ is the summation over $E$.
Let:
:$F'= F \cap J$
From Set Difference and Intersection form Partition:
:$F = F' \cup F \setminus J$
Let $E' \subseteq J$:
:$E' \supseteq F'$
We have:
:$E' \cap F \setminus J = \O$
Let:
:$E = E' \cup F \setminus J$
From Set Union Preserves Subsets:
:$E \supseteq F$
From $(1)$:
:$\ds \sum_{i \mathop \in E} g_i \in U$
==== Case : $F \setminus J = \O$ ====
Let:
:$F \setminus J = \O$
From Union with Empty Set:
:$E = E'$
Hence:
:$\ds \sum_{i \mathop \in E'} g_i \in U$
==== Case : $F \setminus J \ne \O$ ====
Let:
:$F \setminus J \ne \O$
We have:
Hence:
:$\ds \sum_{i \mathop \in E'} g_i \in U$
In either case:
:$\ds \sum_{i \mathop \in E'} g_i \in U$
Since $U$ was arbitrary, it follows that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$ by definition.
Category:Generalized Sum Restricted to Non-zero Summands
Generalized Sum Restricted to Non-zero Summands/Sufficient Condition

== Theorem ==
Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g }_{i \in I}$ be an indexed family of elements of $G$.
Let $J = \set{i \in I : g_i \ne 0_G}$
Let $h \in G$.
Let the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converge to $h$.
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$.
== Proof ==
Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.
By definition of convergent net:
:$(2) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' \implies \ds \sum_{j \mathop \in E'} g_j \in U$ 
where $\ds \sum_{j \mathop \in E'} g_j$ is the summation over $E$.
We have:
:$F' \subseteq J \subseteq I$.
Let $E \subseteq I$:
:$E \supseteq F'$
Let:
:$E' = E \cap J$
From Set Intersection Preserves Subsets and Intersection with Subset is Subset:
:$E' \supseteq F'$
From $(2)$:
:$\ds \sum_{j \mathop \in E'} g_j \in U$
From Set Difference Union Intersection:
:$E = E' \cup E \setminus J$
From Set Difference and Intersection are Disjoint:
:$E' \cap E \setminus J = \O$
==== Case : $E \setminus J = \O$ ====
Let:
:$E \setminus J = \O$
From Union with Empty Set:
:$E = E'$
Hence:
:$\ds \sum_{i \mathop \in E} g_i \in U$
==== Case : $E \setminus J \ne \O$ ====
Let:
:$E \setminus J \ne \O$
We have:
Hence:
:$\ds \sum_{i \mathop \in E} g_i \in U$
In either case:
:$\ds \sum_{i \mathop \in E} g_i \in U$
Since $U$ was arbitrary, it follows that $\ds \sum_{i \mathop \in I} g_i$ converges to $h$ by definition.
Category:Generalized Sum Restricted to Non-zero Summands
Generalized Sum is Linear

== Theorem ==
Let $\family {z_i}_{i \mathop \in I}$ and $\family {w_i}_{i \mathop \in I}$ be $I$-indexed families of complex numbers.
That is, let $z_i, w_i \in \C$ for all $i \in I$.
Let $\ds \sum \set {z_i: i \in I}$ and $\sum \set {w_i: i \mathop \in I}$ converge to $z, w \in \C$, respectively.
Then:
:$(1): \quad \ds \sum \set {z_i + w_i: i \in I}$ converges to $z + w$
:$(2): \quad \forall \lambda \in \C: \ds \sum \set {\lambda z_i: i \in I}$ converges to $\lambda z$
== Proof ==
=== Proof of $(1)$ ===
Let $\epsilon > 0$.
To verify the convergence, it is necessary to find a finite $F \subseteq I$ such that:
:$\ds \map d {\sum_{i \mathop\in G} z_i + w_i, z + w} < \epsilon$ for all finite $G$ with $F \subseteq G \subseteq I$
Now let $F_z, F_w \subseteq I$ be finite subsets of $I$ such that:
:$\ds \map d {\sum_{i \mathop \in G} z_i, z} < \frac \epsilon 2$ for all finite $G$ with $F_z \subseteq G \subseteq I$
:$\ds \map d {\sum_{i \mathop \in G} w_i, w} < \frac \epsilon 2$ for all finite $G$ with $F_w \subseteq G \subseteq I$
The set $F_z \cup F_w$ will be the sought $F$. Let $G$ be finite such that $F_z \cup F_w \subseteq G \subseteq I$.
It follows that:
From the definition of convergence, it is concluded that:
:$\ds \sum \set {z_i + w_i: i \in I} = z + w$
=== Proof of $(2)$ ===
Let $\epsilon > 0$.
To verify the convergence, it is necessary to find a finite $F \subseteq I$ such that:
:$\ds \map d {\sum_{i \mathop\in G} \lambda z_i, \lambda z} < \epsilon$ for all finite $G$ with $F \subseteq G \subseteq I$
Now let $F_z \subseteq I$ be a finite subset of $I$ such that:
:$\ds \map d {\sum_{i \mathop \in G} z_i, z} < \frac \epsilon {\size \lambda}$
for all finite $G$ with $F_z \subseteq G \subseteq I$
The set $F_z$ will be the sought $F$.
Let $G$ be finite such that $F_z \subseteq G \subseteq I$.
It follows that:
From the definition of convergence, conclude that:
:$\ds \sum \set {\lambda z_i: i \in I} = \lambda z$
== Also see ==
* Convergence of Generalized Sum of Complex Numbers establishes a partial converse.
Category:Generalized Sums
Category:Complex Numbers
Generalized Sum is Monotone

== Theorem ==
Let $I$ be an indexing set.
Let $\family {a_i}_{i \mathop \in I}$ be an $I$-indexed family of positive real numbers.
That is, let $a_i \in \R_{\ge 0}$ for all $i \in I$.
Then, for every finite subset $F$ of $I$:
:$\ds \sum_{i \mathop \in F} a_i \le \sum_{i \mathop \in I} a_i$
provided the generalized sum on the right converges.
== Proof ==
Let $\struct {\FF, \subseteq}$ be the set of finite subsets of $I$, ordered by inclusion.
From Finite Subsets form Directed Set, $\struct {\FF, \subseteq}$ is directed.
For brevity write:
:$\ds s_X = \sum_{i \mathop \in X} a_i$ 
for $X \in \FF$. 
 that:
:$\ds \sum_{i \mathop \in F} a_i > \sum_{i \mathop \in I} a_i$
for some finite subset $F$ of $I$.
Take $\epsilon > 0$ such that:
:$\ds \sum_{i \mathop \in F} a_i > \sum_{i \mathop \in I} a_i + \epsilon$ 
Since $\family {s_X}_{X \mathop \in \FF}$ converges to $\ds \sum_{i \mathop \in I} a_i$, there exists a finite set $G$ such that if $X$ is a finite set with $X \supseteq G$, we have:
:$\ds \size {\sum_{i \mathop \in X} a_i - \sum_{i \mathop \in I} a_i} < \epsilon$ 
Then in particular for $X = F \cup G$ we have:
:$\ds \sum_{i \mathop \in F \cup G} a_i \le \sum_{i \mathop \in I} a_i + \epsilon < \sum_{i \mathop \in F} a_i$ 
However since $F \subseteq F \cup G$ and $a_i \in \R_{\ge 0}$ for each $i \in F \cup G$, we have:
:$\ds \sum_{i \mathop \in F} a_i \le \sum_{i \mathop \in F \cup G} a_i < \sum_{i \mathop \in F} a_i$
a contradiction.
So:
:$\ds \sum_{i \mathop \in F} a_i \le \sum_{i \mathop \in I} a_i$
for each $F \in \FF$.
Category:Generalized Sums
Generalized Sum of Constant Zero Converges to Zero

== Theorem ==
Let $G$ be a commutative topological semigroup with identity $0_G$.
Let $\family{g_i}_{i \in I}$ be the indexed family of $G$ defined by:
:$\forall i \in I : g_i = 0_G$
Then:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $0_G$
== Proof ==
Let $\FF$ denote the set of finite subsets of $I$.
From Power of Identity is Identity:
:$\forall F \in \FF : \ds \map \phi F = \sum_{i \mathop \in F} g_i = 0_G$
Hence the net $\ds \sum \set {g_i: i \in I}$ is a constant mapping.
From Constant Net is Convergent:
:the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $0_G$
Category:Generalized Sums
Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent

== Theorem ==
Let $V$ be a Banach space.
Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.
Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.
Let $J \subseteq I$.
Then:
:the generalized sum $\ds \sum_{j \mathop \in J} v_j$ is absolutely net convergent.
== Proof ==
By definition of absolute net convergence, let:
:$\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq J$ be finite.
From Subset Relation is Transitive:
:$F \subseteq I$
From Absolutely Convergent Generalized Sum Converges to Supremum:
:$\ds \sum_{j \mathop \in F} \norm{v_j} \le M$
Since $F \subseteq J$ was arbitrary, it follows that:
:$\forall F \subseteq J : F$ is finite $: \ds \sum_{j \mathop \in F} \norm{v_j} \le M$
From Bounded Generalized Sum is Absolutely Convergent:
:$\ds \sum_{j \mathop \in J} \norm{v_j}$ is absolutely net convergent
Category:Banach Spaces
Category:Generalized Sums