Sequences of Projections in 2-Sequence Space do not Converge in Uniform 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.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

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 $\sequence {P_n}_{n \mathop \in \N}$ does not converge to $I$ in the uniform operator topology.

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

By definition:


 * $\forall \epsilon \in \R_{> 0} : \exists N \in \N : \forall n \in \N : n > N \implies \norm {I - P_n} < \epsilon$

Suppose $\mathbf e_{N + 1} \in \ell^2$ is such that:


 * $\mathbf e_{N + 1} = \tuple {\underbrace {0, \ldots, 0}_N, 1, 0 \ldots}$

Then:

This has to hold for all $\epsilon > 0$.

However, for $\epsilon < 1$ this does not hold.

Hence, we have a contradiction.

Therefore, $\sequence {P_n}_{n \mathop \in \N}$ does not converge to $I$ in the uniform operator topology.