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)