Is compactness a stronger form of continuity?

Question

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has got a convergent subsequence. We can characterize compactness in terms of weak convergence: $T$ is compact if and only if it maps weakly convergent sequences into norm convergent ones.

Up to now I would have casually said:

$T$ is compact iff $T$ is continuous when its domain is equipped with weak topology and its range with norm topology.

and I would not have been alone: I've heard this from more than one of my professors. Well, this turns out to be false, as I read on Problem VI.34 of Reed & Simon's Methods of modern mathematical physics:

Show that in a Hilbert space $H$, a map $T \colon H \to H$ is continuous when its domain is given the weak topology and its range the norm topology if and only if $T$ has finite rank!

(the exclamation point is part of the original text).

At the moment this problem is beyond my grasp, since I know almost nothing of the necessary topological tools (that is - nets, Moore-Smith convergence, and the like). But I'm curious about this and wanted to share my curiosity with the community.

Has somebody got any idea on how to prove this, or even some hint that could help me to catch this intuitively?

Answer 1

There is the following characterization of compactness:

An operator $T: H \to H$ is compact if and only if $T|_{B} : B \to H$ is continuous, where $B$ is the closed unit ball of $H$ equipped with the weak topology and $H$ is equipped with its usual norm topology.

You can find a proof of this fact e.g. in Pedersen's Analysis Now as part of Theorem 3.3.3. However, this result relies heavily on net techniques, but you could take the opportunity to learn about that in Chapter 1 of that book.

To see that continuity implies compactness in your sense, simply observe that continuity implies that $T(B)$ is compact, and as $T(B)$ is metrizable, every sequence in $B$ has a subsequence whose image under $T$ converges. The other direction is a bit more complicated but not really difficult.

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Now suppose that $T: H \to H$ is weak-norm continuous. Then the seminorm on $H$ given by $x \mapsto \|Tx\|$ is continuous as a map $(H, \text{weak}) \to \mathbb{R}$, so by a standard result on locally convex spaces and the definition of the weak topology (see also Tim's answer to your question), there are points $x_{1},\ldots,x_{n} \in H$ such that $\|Tx\| \leq \max{\{|\langle x,x_{i}\rangle|\,:\,i = 1,\ldots,n\}}$ for all $x \in H$. But this means that $T$ must vanish on the orthogonal complement of the span of $x_1, \ldots, x_n$, so certainly $T$ has finite rank.

Answer 2

First you'll need to understand that a basis of the weak topology consists of the sets $$ V_{\epsilon, y_1, ..., y_n} = \{x \in H: |(x, y_i)| \lt \epsilon \} $$ for any $\epsilon \gt 0$ and a finite set of vectors $y_i$. To be more precise: These sets form a neighborhood basis at zero, which is what we'll need for the proof.

Now, if a linear transformation $$ T: H_{weak} \to H_{norm} $$ is continuous, then the inverse image of the open unit ball in $H_{norm}$ has to contain such a basic weak neighborhood $V_{\epsilon, y_1, ..., y_n}$. If H is infinite dimensional, we can find an $x$ such that $(x, y_i) = 0$ for $i = 1,..., n$. This implies that $k x \in V$ for any $k$, so that $T x = 0$ needs to hold (which implies that the operator norm of $T$ has to be zero).

We may assume that the $y_i$ are orthonormal (why?). Given any $x \in H$, we can write $$ z = x - \sum_{i = 1}^{n} (x, y_i) \; y_i $$ Now we know that $A z = 0$ by construction, which means that we know $T x$ for any arbitrary $x$, it is $$ T x = \sum_{i = 1}^{n} (x, y_i) \; y_i $$ So much for your concrete question.

Infinite dimensional normed spaces are fundamentally different from finite normed spaces, for example:

• The unit ball is compact iff the space is finite dimensional.

Therefore the compact operators are of interest because they retain some properties from the finite dimensional case in infinte dimensions, as is the weak operator topology, that makes them interesting.

BTW: I think your professors were talking about the fact that compact operators are those that are continuous from the weak unit ball of $H_{weak}$ to $H_{norm}$. The restriction to the unit ball is essential, and looking at the proof above you can see which step does not work anymore if we add remove this restriction.