1. Let H be a complex inner space. Prove the polar identity
(x, y) =
kx + yk2 − kx − yk2 + ikx + iyk2 − ikx − iyk2
.
4
Show that a norm on a complex vector space is defined by a scalar product if and only
if it satisfies the parallelogram law.
2. A bounded continuous function f : R → C is called almost periodic if for any ε > 0
there exists R > 0 such that for any interval I ⊂ R of length R there exists t ∈ I such that
sup |f (τ + t) − f (τ )| < ε.
τ ∈R
Show that a linear combination of two continuous periodic functions is almost periodic.
3. Prove that the spaces c0 and `p , 1 ≤ p < ∞, are separable, while `∞ is not.
4. Let x, y be linearly independent vectors in an inner space H, kxk = kyk = 1, λ ∈ (0, 1).
Prove that kλx + (1 − λ)yk < 1.
Show that `∞ does not have this property.