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I am currently reading a proof on characterizations of a compact operator for Hilbert spaces. Let $\mathscr{B}_1$ be the closed unit ball of the Hilbert space $H$. One of the equivalent statement involves: $T\in B(H)$ is the norm-limit of finite rank operators if and only if $T_{|\mathscr{B}_1}:\mathscr{B}_1\to H$ is continuous as a function from $\mathscr{B}_1$ endowed with weak topology to $H$ endowed with the norm topology.

The author then proceeds to prove the $\implies$ direction of the result basically by showing that $\{x_n\}$ converges weakly to $x$ in $\mathscr{B}_1$ implies that $\{Tx_n\}$ converges to $Tx$ in norm of $H.$ I know this is the sequential criterion for continuity, but this is only valid if $\mathscr{B}_1$ is a (weak) sequential space. Is it?

I've tried looking for other sources, but to no avail.

Kurome
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1 Answers1

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This is just from the definition of weak convergence. First, suppose $Tx = \phi(x) y$ is an operator of rank $1$, where $\phi \in H^*, y \in H$. Let $(x_\alpha)_\alpha$ be a net in $\mathscr{B}_1$ converging weakly to $x \in \mathscr{B}_1$. Then $$\|Tx - T x_\alpha \| = | \phi(x - x_\alpha) | \|y \| \to 0 .$$ We can then extend this fact to an operator of finite rank by finite additivity. Now if $S$ is an arbitrary compact operator, then if $T$ is a finite-rank operator that norm-approximates $S$, then \begin{align*} \| Sx - S x_\alpha \| & \leq \| Sx - Tx \| + \| Tx - T x_\alpha \| + \| T x_\alpha - S x_\alpha \| \\ & \leq \| S - T \| \|x \| + \| Tx - T x_\alpha \| + \| T - S \| \|x_\alpha \| \\ & \leq 2 \| S - T \| + \| Tx - T x_\alpha \| . \end{align*} Now fix $\epsilon > 0$, and choose $T$ finite-rank so $\| S - T \| < \epsilon / 3$. Then choose $\beta$ so $\alpha \geq \beta \Rightarrow \| Tx - T x_\alpha \| < \epsilon / 3$. Then if $\alpha \geq \beta$, then $\| S x - S x_\alpha \| < \epsilon$.

Therefore $S$ is continuous from the weak $\mathscr{B}_1$ to the normed $H$.

AJY
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  • Does the sequential continuity, as what you have shown imply that $T$ then is continuous wrt to the appropriate topologies?

    Edit: I am not really asking how to show sequential continuity, but I am asking how showing it would imply actual continuity of $T$.

     – Kurome Jul 17 '20 at 16:28
  • @Kurome Are you familiar with nets? A net is a generalization of a sequence such that a function $f : X \to Y$ between any topological spaces is continuous iff for every net $(x_\alpha)\alpha$ in $X$ converging to a point $x \in X$, the net $(f(x\alpha))\alpha$ in $Y$ converges to $f(x)$. I claim you could replace all the sequences $(x_n)$ in my answer with nets $(x\alpha)$ and things would work out just the same. – AJY Jul 17 '20 at 16:43
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    I am! Thanks! I have actually forgotten about that very important difference between nets and sequences! – Kurome Jul 17 '20 at 16:45
  • @Kurome I’ve edited my answer to hopefully cut down on the confusion. – AJY Jul 17 '20 at 16:48