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

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:

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.