Theorem: Pitt's Theorem

Let $\GF \in \set {\R, \C}$.
Let $p, r \in \hointr 1 \infty$ be such that $p < r$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ and $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ be the $p$ and $r$-sequence spaces. 
Let $X$ be a closed vector subspace of $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$.
Let $\struct {\map \BB {X, \map {\ell_p} \GF}, \norm {\, \cdot \,}_{\map \BB {X, \map {\ell_p} \GF} } }$ be the space of bounded linear transformations $X \to \map {\ell_p} \GF$.
Let $\map \KK {X, \map {\ell_p} \GF}$ be the space of compact linear transformations $X \to \map {\ell_p} \GF$.
Let $T : X \to \map {\ell_p} \GF$ be a bounded linear transformation.
Then $T$ is compact.
Hence, $\map \BB {X, \map {\ell_p} \GF} = \map \KK {X, \map {\ell_p} \GF}$.

Proof:
Let $w$ denote the weak topology on $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$.
From P-Sequence Space is Reflexive for Finite p Greater Than One:
:$\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ is reflexive. 
From Closed Vector Subspace of Reflexive Banach Space is Reflexive:
:$X$ is reflexive.
Let $B_X^-$ be the closed unit ball of $X$.
Hence from Kakutani's Theorem:
:$\struct {B_X^-, w}$ is compact.
From Weakly Compact Set in P-Sequence Space is Weakly Metrizable, it follows that:
:$\struct {B_X^-, w}$ is metrizable. 
We now show that $T \restriction_{B_X^-} : \struct {B_X^-, w} \to \struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is continuous.
From Continuous Image of Compact Space is Compact, we will then have that $T \sqbrk {B_X^-}$ is compact in $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Since both $\struct {B_X^-, w}$ and $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ are metrizable, we can apply Sequential Continuity is Equivalent to Continuity in Metric Space for this purpose.
We therefore show that if $\sequence {x_n}_{n \mathop \in \N}$ is a sequence in $B_X^-$ that converges weakly to $x \in B_X^-$, then $\norm {T x_n - T x}_p \to 0$.
By replacing $x_n$ with $x_n - x$, it is enough to show that if $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to ${\mathbf 0}_X$, then $\norm {T x_n}_p \to 0$ by Linear Combination of Weakly Convergent Sequences is Weakly Convergent.
If $\norm {x_n}_{\ell_r} \to 0$, then $\norm {T x_n}_p \to 0$ from the boundedness of $T$. 
{{AimForCont}} there exists $\sequence {x_n}_{n \mathop \in \N}$ converges weakly (but not in norm) to ${\mathbf 0}_X$ such that $\norm {T x_n}_p \not \to 0$.
Since $\norm {T x_n}_p \not \to 0$, there exists $m_2 > 0$ such that:
:$\norm {T x_n}_p \ge m_2$ for infinitely many $n \in \N$.
Hence passing to a subsequence $\sequence {x_{n_j} }_{j \mathop \in \N}$, we can assume that:
:$\norm {T x_{n_j} }_p \ge m_2$ for each $n \in \N$.
If $\sequence {x_{n_j} }_{j \mathop \in \N}$ in norm to ${\mathbf 0}_X$, then we have $\norm {T x_{n_j} }_p \to 0 < m_2$ by boundedness, a contradiction. 
Hence suppose that $\sequence {x_{n_j} }_{j \mathop \in \N}$ does not converge in norm to ${\mathbf 0}_X$.
Hence there exists $m_1 > 0$ such that:
:$\norm {x_{n_j} }_r \ge m_1$ for infinitely many $n \in \N$.
By passing to a subsequence, we can assume that:
:$\norm {x_{n_j} }_r \ge m_1$ for each $n \in \N$.
Relabel $y_j = x_{n_j}$ for each $j \in \N$.
Then we possess a sequence $\sequence {y_j}_{j \mathop \in \N}$ in $X$ such that:
:$\norm {y_j}_r \ge m_1$
and:
:$\norm {T y_j}_p \ge m_2$ 
From Limit of Subsequence equals Limit of Sequence, $\sequence {y_j}_{j \mathop \in \N}$ converges weakly to ${\mathbf 0}_X$.
Define:
:$\ds z_j = \frac {y_j} {\norm {y_j}_r}$ for each $j \in \N$.
By {{NormAxiomVector|1}}, we have $\norm {z_j}_r = 1$ for each $j \in \N$. 
Since:
:$\ds \frac 1 {\norm {y_j}_r} \le \frac 1 {m_1}$ 
Hence from Weakly Null Sequence in Normed Vector Space Multiplied by Bounded Sequence is Weakly Null, $\sequence {z_j}_{j \mathop \in \N}$ converges weakly to ${\mathbf 0}_X$.
Define $e_n : \N \to \GF$ by:
:$\ds \map {e_n} k = \begin{cases}1 & n = k \\ 0 & \text{otherwise}\end{cases}$ 
From P-Sequence Space admits Schauder Basis, $\sequence {e_n}_{n \mathop \in \N}$ is a Schauder basis for both $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ and $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$. 
From Normalized Sequence in P-Sequence Space which Vanishes Pointwise has Basic Subsequence Equivalent to Schauder Basis: Corollary:
:there exists a subsequence of $\sequence {z_n}_{n \mathop \in \N}$ that is basic and equivalent to $\sequence {e_n}_{n \mathop \in \N}$ in $\map {\ell_r} \GF$.
Pass to this subsequence.
From Limit of Subsequence equals Limit of Sequence, $\sequence {z_n}_{n \mathop \in \N}$ still converges weakly to ${\mathbf 0}_X$.
From Characterization of Norm-to-Weak and Weak-to-Weak Continuity of Linear Transformations, $T$ is continuous as a mapping $\struct {X, w} \to \struct {\map {\ell_p} \GF, w}$. 
Hence $\sequence {T z_n}_{n \mathop \in \N}$ converges weakly to ${\mathbf 0}_{\ell_p}$.
Since $\norm {T z_n}_p \ge m_2$ for each $n \in \N$, we have that:
:$\ds \sequence {\frac {T z_n} {\norm {T z_n}_p} }_{n \mathop \in \N}$ converges weakly to ${\mathbf 0}_{\ell_p}$.
while:
:$\ds \norm {\frac {T z_n} {\norm {T z_n}_p} }_p = 1$ 
Then from Normalized Sequence in P-Sequence Space which Vanishes Pointwise has Basic Subsequence Equivalent to Schauder Basis: Corollary:
:there exists a subsequence of $\ds \sequence {\frac {T z_n} {\norm {T z_n}_p} }_{n \mathop \in \N}$ that is basic and equivalent to $\sequence {e_n}_{n \mathop \in \N}$ in $\map {\ell_p} \GF$.
From Weakly Convergent Sequence in Normed Vector Space is Bounded, there exists $M_2 > 0$ such that:
:$\norm {T z_n}_p \le M_2$
Hence:
:$\ds \frac 1 {M_2} \le \norm {T z_n}_p \le \frac 1 {m_2}$ for each $n \in \N$.
From Equivalence of Basic Sequences in Normed Vector Spaces is Preserved by Scalar Multiplication, we have that: 
:$\sequence {T z_n}_{n \mathop \in \N}$ is equivalent to $\sequence {\norm {T z_n}_p e_n}_{n \mathop \in \N}$ in $\map {\ell_p} \GF$.
From Standard Schauder Basis in P-Sequence Space Multiplied by Bounded Sequence is Equivalent to Original Basis, we have that:
:$\sequence {\norm {T z_n}_p e_n}_{n \mathop \in \N}$ is equivalent to $\sequence {e_n}_{n \mathop \in \N}$ in $\map {\ell_p} \GF$.
From Equivalence of Basic Sequences in Banach Spaces is Transitive, we have that:
:$\sequence {T z_n}_{n \mathop \in \N}$ is equivalent to $\sequence {e_n}_{n \mathop \in \N}$ in $\map {\ell_p} \GF$.
Hence there exists $C_2 > 0$ such that for each $\xi \in \map {c_{00} } \GF$, we have that:
:$\ds \norm {\sum_{j \mathop = 1}^\infty \map \xi j T z_j}_p \ge C_2 \norm {\sum_{j \mathop = 1}^\infty \map \xi j e_j}_p$
from Basic Sequences in Banach Spaces are Equivalent iff Map between is Linear Isomorphism: Corollary, where $\map {c_{00} } \GF$ denotes the space of almost-zero sequences. 
We recall that:
:$\sequence {z_n}_{n \mathop \in \N}$ is equivalent to $\sequence {e_n}_{n \mathop \in \N} \in \map {\ell_r} \GF$.
Hence by Basic Sequences in Banach Spaces are Equivalent iff Map between is Linear Isomorphism: Corollary, there exists $C_1 > 0$ such that:
:$\ds \norm {\sum_{j \mathop = 1}^\infty \map \xi j z_j}_r \le C_1 \norm {\sum_{j \mathop = 1}^\infty \map \xi j e_j}_r$
for each $\xi \in \map {c_{00} } \GF$.
Let $n \in \N$ and define $\xi_n : \N \to \GF$ by:
:$\ds \map {\xi_n} j = \begin{cases}1 & j \le n \\ 0 & \text{otherwise}\end{cases}$
Taking $\xi = \xi_n$ in the previous two inequalities, we obtain:
{{begin-eqn}}
{{eqn	| l = \norm {\sum_{j \mathop = 1}^n z_j}_r
		| r = \norm {\sum_{j \mathop = 1}^\infty \map {\xi_n} j z_j}_r 
}}
{{eqn	| o = \le
		| r = C_1 \norm {\sum_{j \mathop = 1}^\infty \map {\xi_n} j e_j}_r
}}
{{eqn	| r = C_1 \norm {\sum_{j \mathop = 1}^n e_j}_r 
}}
{{eqn	| r = C_1 \paren {\sum_{j \mathop = 1}^n 1^r}^{1/r} 
}}
{{eqn	| r = C_1 n^{1/r}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn	| l = \norm {\sum_{j \mathop = 1}^n T z_j}_p
		| r = \norm {\sum_{j \mathop = 1}^\infty \map {\xi_n} j T z_j}_p
}}
{{eqn	| o = \ge
		| r = C_2 \norm {\sum_{j \mathop = 1}^\infty \map {\xi_n} j e_j}_p 
}}
{{eqn	| r = C_2 \norm {\sum_{j \mathop = 1}^n e_j}_p 
}}
{{eqn	| r = C_2 \paren {\sum_{j \mathop = 1}^n 1^p}^{1/p}
}}
{{eqn	| r = C_2 n^{1/p}
}}
{{end-eqn}}
We then have:
{{begin-eqn}}
{{eqn	| l = C_2 n^{1/p}
		| o = \le
		| r = \norm {\sum_{j \mathop = 1}^n T z_j}_p
}}
{{eqn	| o = \le
		| r = \norm {\map T {\sum_{j \mathop = 1}^n z_j} }_p 
		| c = $T$ is linear
}}
{{eqn	| o = \le
		| r = \norm T_{\map \BB {X, \map {\ell_p} \GF} } \norm {\sum_{j \mathop = 1}^n z_j}_p 
		| c = Fundamental Property of Norm on Bounded Linear Transformation
}}
{{eqn	| o = \le
		| r = C_1 \norm T_{\map \BB {X, \map {\ell_p} \GF} } n^{1/r} 
}}
{{end-eqn}}
Hence we have:
:$\ds n^{1/r - 1/p} \ge \frac {C_1 \norm T_{\map \BB {X, \map {\ell_p} \GF} } } {C_2}$ for all $n \in \N$.
However, since $p < r$, we have:
:$\ds \frac 1 r - \frac 1 p < 0$ 
Hence taking $n \to \infty$, we obtain:
:$\ds n^{1/r - 1/p} \to 0$
Hence for large enough $n \in \N$, we have:
:$\ds n^{1/r - 1/p} < \frac {C_1 \norm T_{\map \BB {X, \map {\ell_p} \GF} } } {C_2}$
a contradiction.
Hence if $\sequence {x_n}_{n \mathop \in \N}$ is a sequence that converges weakly to ${\mathbf 0}_X$, then $\norm {T x_n}_p \to 0$.
Hence $T \restriction_{B_X^-} : \struct {B_X^-, w} \to \struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is continuous, since $T$ is linear.
Since $\struct {B_X^-, w}$ is compact, we have that $T \sqbrk {B_X^-}$ is compact in $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$. 
From Compact Subspace of Hausdorff Space is Closed, it follows that $T \sqbrk {B_X^-}$ is closed in $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
From Set is Closed iff Equals Topological Closure, we have $T \sqbrk {B_X^-} = \map \cl {T \sqbrk {B_X^-} }$. 
Hence $\map \cl {T \sqbrk {B_X^-} }$ is compact.
Hence $T$ is compact.
{{qed}}
{{Namedfor|Harry Raymond Pitt|cat = Pitt}}