Homework 4, Undergraduate Analysis

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Homework 4, Undergraduate Analysis

Refer to class notes for notions used in this homework. H will be used to denote a Hilbert space and k · k is the norm induced by inner product < ·, · >. The metric on H is given by this norm.

1. Prove that

k · k : H → K is continuous.

2. Given two Hilbert spaces H1 and H2, define product norm on H1× H2 by

k(x₁, x₂)k = max(kx₁kH₁, kx₂kH₂).

(a) Prove that this norm is equivalent to k(x1, x2)k⁰ =

q

kx1k²_H1 + kx2k²_H2.

(b) Prove that vector space addition

+ : H × H → H is continuous with respect to product norm.

(c) Prove the scalor multiplication

· : K × H → H is continuous with respect to product norm.

3. Prove the continuity of inner product: If x_n→ x and y_n → y then < x_n.y_n>→< x, y >.

4. Prove the parallelogram law stated in class.

5. Prove the Pythagorean’s Theorem stated in class.

6. Let E ⊂ H. Prove that (E^⊥)^⊥ is the smallest closed subspace containing E. That is, if F is a closed subspace containing E, then (E^⊥)^⊥⊂ F .

7. Here we study a vector space not isomorphic to its dual. Define

E = R^(N):= {{a_i}^∞_i=1| a_i ∈ R and ∃N such that ai = 0∀i > N } and

V = R^N:= {{a_i}^∞_i=1| a_i ∈ R}.

Clearly, the set {e_i}^∞_i=1, where e_i is the sequence with a_i = 1 and a_j = 0 ∀j 6= i is a basis for E.

(a) Prove that E is not isomorphic to V .

(b) Prove that V is isomorphic to E^∗ and conclude that E is not isomorphic to E^∗. 8. Prove that for any A 6= ∅, l²(A) is complete.

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