1. Let X be a Banach space, f : X × X → C a bilinear form that is separately continuous in
each variable. Show that there exists C > 0 such that
|f (x, y)| ≤ Ckxkkyk
for all x, y ∈ X. Conclude that f is continuous.
2. Let M be a compact metric space, f : M × M → C a function that is separately continuous
in each variable. Show that the set of continuity points of f is dense in M × M .
3. Show that the set
{f ∈ C[0, 1] | f 0 (x) exists for some x ∈ [0, 1]}
is a set of first category in C[0, 1]. Hint: this set is the union of the sets
An = {f ∈ C[0, 1] | there exists x ∈ [0, 1] such that |f (x) − f (y)| ≤ n|x − y| for all y ∈ [0, 1]}.
4. A map F : M1 → M2 between metric spaces is called Lipschitz if there exists L > 0 such that
d(F (x), F (y)) ≤ Ld(x, y)
for all x, y ∈ M1 .
Show that if M is a metric space, X a Banach space and F : M → X a map such that f ◦ F is
Lipschitz for all f ∈ X ∗ , then F is Lipschitz.
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