Functional Analysis Answers

(i) Let V be a Banach space. Prove that if V* separable then V is not separable.

(ii) Give an example of separable Banach space V which has a non-separable dual space V* . 

Let c0 be the space of sequences of complex numbers which converge to 0. That is

c_0 = {(x_i)_i∈N : xi ∈ C, x_i → 0}.

(i) Show that c0 is a closed subspace of L^∞.

(ii) Define a mapping T by

T : L^∞ → c_0

(x_n)_n → (x_n/ n ) _n

Show that T is a (linear bounded) operator. Show that ran T is not closed.

• Prove that C.V100√n-1

Define f(x) = sinx on [0, 2pi]. Find two increasing functions h and g for which f = h — g on 

[0, 2pi]. 

Show that f(x)=2/x+1 is indected

show that f°f=2f

f=x+1

Let H be a Hilbert space.

(i) Let S ⊆ H be any non-empty subset of S. Show that S ^⊥ is a subspace of H.

(ii) Let L ⊆ H be a linear manifold. Show that L^ ⊥⊥ = L

(i) Let V be a Banach space and let L ⊆ V be a linear manifold. Show that L, the closure of L, is a subspace of V .

(ii) Let H be a Hilbert space. Let B be an orthonormal set in H. Show that span B is dense in H if and only if B is an orthonormal basis.

Prove that every (non-zero) Hilbert space H has an orthonormal basis.