Is there a sequence of linear maps which converges pointwise but not in norm?

Question

Does it exist a sequence $(M_n) \in \mathcal{L}(X,Y)$, where $X$ and $Y$ are Banach Spaces, such that it converges pointwise but not in norm?

Answer 1

Yes, there is such a sequence. Let $X = l_1$, $Y = \Bbb{R}$, and $$f_n(x) = \sum_{m = n}^\infty x_m$$ Clearly $f_n$ is linear. If $x = (x_m) \in B_{l_1}$, then $$|f_n(x)| \le \sum_{m = n}^\infty |x_m| \le \sum_{m = 1}^\infty |x_m| \le 1,$$ hence $f_n$ is bounded, with $\|f_n\| \le 1$.

Let's show $f_n \to 0$ pointwise. Fix $x = (x_m)\in l_1$. Because $(x_m)$ is (absolutely) summable, for any $\varepsilon > 0$, we can find some $N$ such that $$n > N \implies \left|\sum_{m=n}^\infty x_m \right| < \varepsilon \implies |f_n(x)| < \varepsilon$$

But, $f_n$ does not converge to $0$ in norm! In particular, take the sequence $(e^n)_{n=1}^\infty$ of points in $l_1$, where $e^n_m = 1$ if $n = m$ and $0$ otherwise. We have $f_n(e^n) = 1$ for all $n$, hence $\|f_n\| = 1$ for all $n$.