91Ó°ÊÓ

Show that a finite lower semicontinuous convex function \(f\) that is defined on a whole Banach space must be continuous.

Prove that every convex function \(f\) defined on an open interval \(I \subset \mathbf{R}\) is differentiable at all but (at most) countably many points of \(I\). Hint: Observe that \(d^{+} f(x)(1)=\lim _{t \rightarrow 0+} \frac{f(x+t)-f(x)}{t}\), the derivative of \(f\) at \(x\) from the right, is a nondecreasing function of \(x\). Prove then that, at any point where \(f\) fails to be differentiable, the monotone function \(x \rightarrow\) \(d^{+} f(x)(1)\) has a jump. Because there are not more than a countable number of jumps, the conclusion follows.

Let \(X\) be a Banach space and let \(f\) be a continuous convex function on \(X^{*}\) that is \(w^{*}\) -lower semicontinuous. Show that if \(f\) is Fréchet differentiable at \(x^{*} \in X^{*}\), then \(f^{\prime}\left(x^{*}\right) \in X\). Hint: The derivative, as a uniform limit of quotients in \(B_{X^{*}}\), is also \(w^{*}\) -lower semicontinuous. Then use its linearity to see that \(f^{\prime}\left(x^{*}\right)\) is a functional that is \(w^{*}\) -continuous on \(B_{X *}\) and apply Theorem \(4.44\).

Find an example of a Gâteaux differentiable norm on a Banach space that is not Fréchet differentiable at some points. Hint: Any equivalent renorming by a Gâteaux differentiable norm of \(\ell_{1}\) (Theorem 8.13) satisfies the requircment (Theorem 8.26).

Show that the norm of \(C[0,1]\) is nowhere Fréchet differentiable. Show that the norm of \(C[0,1]\) is Gâteaux differentiable at \(x \in S_{C[0,1]}\) if and only if \(|x|\) attains its maximum at exactly one point of \([0,1]\). Hint: Note that the distance between two different Dirac measures in \(C[0,1]^{*}\) is two. Given \(x \in S_{C[0,1]}\), choose \(t_{0} \in[0,1]\) such that \(x\left(t_{0}\right)=1\). Then choose \(t_{n} \neq t_{0}\) such that \(x\left(t_{n}\right) \rightarrow 1\). By the Šmulian lemma, \(x\) is not a point of Fréchet differentiability of the supremum norm on \(C[0,1]\). For the second part, assume that \(x \in S_{C[0,1]}\) is such that \(x\left(t_{0}\right)=1\) and \(|x(t)|<1\) for every \(t \neq t_{0} .\) Put \(H=\left\\{f \in C[0,1]^{*} ;\|f\| \leq 1, f(x)=1\right\\} .\) If \(H \cap B_{C[0,1] *} \neq\left\\{\delta_{t_{0}}\right\\}\), then this intersection would have at least two extreme points that would be extreme points of \(B_{C[0,1]^{*}}\). All the extreme points of \(B_{C[0,1]^{*}}\) are \(\pm\) Dirac measures (Lemma 3.42).

Show that the nonlinear operator \(\varphi\) from \(L_{2}[0,1]\) into \(L_{2}[0,1]\) defined by \(\varphi(x): t \mapsto \sin (x(t))\) is everywhere Gâteaux but nowhere Fréchet differentiable.

A closed convex bounded set \(C\) in a Banach space \(X\) is said to have normal structure if every closed convex subset \(K\) of \(C\) containing more than one point has a non-diametral point in \(K .\) Show that every compact convex set in a Banach space has normal structure.

Show that every exposed point of a convex set is extreme and give an example of an extreme point that is not exposed.

Show that none of the spaces \(C[0,1], c_{0}\), or \(L_{1}[0,1]\) is isomorphic to a dual space.

Let \(C=\left\\{x \in B_{\ell_{1}} ; x_{i} \geq 0\right\\}\). Show that \(C\) is not a closed convex hull of its \(w^{*}\) -strongly exposed points. A strongly exposed point of \(C \subset X^{*}\) is called \(w^{*}\) -strongly exposed if it is strongly exposed by a functional from \(X\).