Sequences of Projections in 2-Sequence Space Converges in Strong Operator Topology

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Theorem

Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $p$-sequence normed vector space.

Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.

For $n \in \N$ let $P_n \in \map {CL} {\ell^2}$ be the projection operator over $\ell^2$.

Let $\sequence {P_n}_{n \mathop \in \N}$ be a sequence.

Let $I \in \map {CL} {\ell^2}$ be the identity operator.


Then $\sequence {P_n}_{n \mathop \in \N}$ converges to $I$ in the strong operator topology.


Proof

Let $\mathbf a = \tuple {a_1, a_2, \ldots} \in \ell^2$.

Then:

\(\ds \norm {I \mathbf a - P_n \mathbf a}^2_2\) \(=\) \(\ds \norm {\ldots, 0, a_{n+1}, a_{n + 2}, \ldots}^2_2\) Definition of Projection Operator over 2-Sequence Spaces
\(\ds \) \(=\) \(\ds \sum_{k \mathop = n + 1}^\infty \size {a_k}^2\)
\(\ds \leadsto \ \ \) \(\ds \lim_{n \mathop \to \infty } \norm {I \mathbf a - P_n \mathbf a}^2_2\) \(=\) \(\ds \sum_{k \mathop = n + 1}^\infty \size {a_k}^2\)
\(\ds \) \(=\) \(\ds 0\) Definition of P-Sequence Space, Tail of Convergent Series tends to Zero

Hence:

$\ds \lim_{n \mathop \to \infty } \norm {I \mathbf a - P_n \mathbf a}_2 = 0$

By definition, $\sequence {P_n}_{n \mathop \in \N}$ converges to $I$ in the strong operator topology.

$\blacksquare$


Sources

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