0.3. The geometry of Hilbert spaces 25
an orthogonal basis uj, 1 j n, such that

j
αjuj
2
2
=

j
|αj|2.
Let
f =

j
αjuj. Then by the triangle and Cauchy–Schwarz inequalities,
f
1

j
|αj|uj
1

j
uj 2
1
f
2
and we can choose m2 =

j
uj 1.
In particular, if fn is convergent with respect to . 2, it is also convergent
with respect to . 1. Thus .
1
is continuous with respect to .
2
and attains
its minimum m 0 on the unit sphere (which is compact by the Heine–Borel
theorem, Theorem 0.12). Now choose m1 = 1/m.
Problem 0.18. Show that the norm in a Hilbert space satisfies f + g =
f + g if and only if f = αg, α 0, or g = 0.
Problem 0.19 (Generalized parallelogram law). Show that, in a Hilbert
space,
1≤jk≤n
xj xk
2
+
1≤j≤n
xj
2
= n
1≤j≤n
xj
2.
The case n = 2 is (0.47).
Problem 0.20. Show that the maximum norm on C[0, 1] does not satisfy
the parallelogram law.
Problem 0.21. In a Banach space, the unit ball is convex by the triangle
inequality. A Banach space X is called uniformly convex if for every
ε 0 there is some δ such that x 1, y 1, and
x+y
2
1 δ imply
x y ε.
Geometrically this implies that if the average of two vectors inside the
closed unit ball is close to the boundary, then they must be close to each
other.
Show that a Hilbert space is uniformly convex and that one can choose
δ(ε) = 1 1
ε2
4
. Draw the unit ball for
R2
for the norms x
1
=
|x1| + |x2|, x
2
= |x1|2 + |x2|2, and x

= max(|x1|, |x2|). Which of
these norms makes
R2
uniformly convex?
(Hint: For the first part, use the parallelogram law.)
Problem 0.22. Suppose Q is a vector space. Let s(f, g) be a sesquilinear
form on Q and q(f) = s(f, f) the associated quadratic form. Prove the
parallelogram law
(0.54) q(f + g) + q(f g) = 2q(f) + 2q(g)
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