Different Types of Continuity in Reflexive Banach Space

Question

Let $X$ be a reflexive Banach space with dual $X^*$. Let $K\subset X$ be a nonempty closed convex set. The mapping $F: K\rightarrow X^*$ is said to be:

• weakly continuous if $F$ is continuous w.r.t. the weak$^*$ topology on $X^*$ and the induced topology on $K$;

• continuous on finite dimensional subspaces if for any subspaces $M\subset X$ the restriction of F to $K\cap M$ is continuous w.r.t the weak$^*$ topology on $X^*$ and the induced topology on $K\cap M$;

• hemicontinuous if for any $x, y\in K$ the restriction of $F$ to $[x, y]$ is continuous w.r.t the weak$^*$ topology on $X^*$ and the induced topology on $[x, y]$.

We are easy to verify that

weak continuity $\Rightarrow$ continuity on finite dimensional subspaces

continuity on finite dimensional subspaces $\Rightarrow$ hemicontinuity.

The reverse is not true in general.

I am stuck in constructing counterexamples for the reverse implication.

Thank you for all comments and helping.

Answer 1

I. Conterexample: continuity on finite dimensional subspaces doesn't imply weak continuity

Let $X = \ell_2$ (or any other infinite dimensional reflexive Banach space) and $K$ be the closed unit ball of $X$. Let $u$ be an arbitrary non-zero vector in $X^*$.

Let $e_{\alpha}$ be a Hamel basis in $X$ with $\|e_{\alpha}\| = 1$. Fix an arbitrary unbounded sequence of real numbers $c_{\alpha}$. Define a map $F$ on vectors $e_{\alpha}$ by $F(e_{\alpha}) = c_{\alpha} u$. Now extend $F$ to $K$ by linearity. We get a map $F$ from $K$ to $X^*$. Since $F$ is linear, it is continuous on every bounded subspace of $X$. On the other hand, since the sequence $c_{\alpha}$ is unbounded, $F$ is unbounded on $K$ (and thus it is not continuous).

II. Conterexample: hemicontinuity doesn't imply continuity on finite dimensional subspaces

Let $X={\mathbb R}^2$, $K$ be the unit ball of $X$, and $u\in X^*\setminus \{0\}$. Define $F(x,y)$ as follows $$F(x) = \cases {0,&if $x=0$;\\ \frac{x_1x_2^2}{x_1^2+x_2^{10}} \cdot u, &otherwise.} $$

It's easy to check that $F$ is continuous on every segment $[x,y]$ but $F$ is not continuous at $0$ (e.g. consider the sequence $(1/k^5, 1/k)$ that tends to $0$ as $k\to\infty$ but $F(1/k^5, 1/k)\not\to 0$).

P.S. Note that since $X$ is reflexive, there is no difference between the weak-* topology and weak topology on $X^*$.