Let $\mathcal H = \ell^2 (\Bbb N)$ and $P_n$ be the projection onto $\text {span}\{e_0,e_1, e_2, \cdots, e_n \}.$ Show that the sequence $\{P_n\}_{n \geq 1}$ does not converge in norm topology.
Attempt $:$ Here for any $x \in \ell^2$ we have $$P_n (x) = \sum\limits_{i=0}^{n} \left \langle e_i, x \right \rangle e_i.$$
So if $P_n \xrightarrow {\text {NORM}} P$ then $P_n \xrightarrow {\text {SOT}} P,$ where SOT denotes the strong operator topology. But then $$P(x) = \sum\limits_{n=0}^{\infty} \left \langle e_n, x \right \rangle e_n$$ The series on the right is the Fourier-Bessel series for $x$ which converges and since $\{e_n\ |\ n \geq 0 \}$ is an orthonormal basis for $\ell^2$ it follows that the Fourier-Bessel series for $x$ converges to $x.$ Hence $Px = x,$ for all $x \in \mathcal H.$ But then $P_n \xrightarrow {\text {NORM}} I,$ where $I$ is the identity operator. But this in turn implies $$\sup\limits_{\|x\| = 1} \sum\limits_{i \gt n} \left \lvert \left \langle e_i, x \right \rangle \right \rvert^2 \longrightarrow 0\ \text {as}\ n \to \infty.$$ Can it happen in any circumstances?
Any help or suggestion in this regard will be highly appreciated. Thanks for your time.
For fixed $n$, choose some $j>n$, and choose $x=e_j$. Then $\|x\|=\|e_j\|=1$, so $\sum_{i>n}|\langle e_i,x\rangle|\geq |\langle e_j,x\rangle|=|\langle e_j,e_j\rangle|=1$. Hence $\sup_{\|x\|=1}\sum_{i>n}|\langle e_i,x\rangle|\geq 1$ independently of $n$. So indeed, we cannot have $\sup_{\|x\|=1}\sum_{i>n}|\langle e_i,x\rangle|\to 0$ if $n\to\infty$.