Math205b Winter Term 2003

Assignment 16

1. Prove that a normed vector space (V, |·|_{V}) is an inner product space iff

|x + y|_{V}^{2} + |x − y|_{V}^{2} = 2(|x|_{V}^{2} + |y|_{V}^{2} ) , x, y ∈ V .

[Hint: polarization identity.]

2. Give an example of a norm which cannot be derived from an inner product and an example of a metric which is not induced by a norm.

3. Define

l_{∞}(𝕂) := {x ∈ 𝕂^{ℕ} | |x|_{∞} := sup_{n∈ℕ}|x_{n}| < ∞}

and

c_{0}(𝕂) := {x ∈ l_{∞}(𝕂) | lim_{n→∞} x_{n} = 0} .

Show that (l_{∞}(𝕂), | · |_{∞}) and (c_{0}(𝕂), | · |_{∞}) are complete normed vector spaces.

4. Two norms |·|_{1} and |·|_{2} on a vector space V are said to be equivalent iff there exists a constant c ≥ 1 such that

\frac{1}{c} |x|_{1} ≤ |x|_{2} ≤ c |x|_{1} , x ∈ V .

Prove that equivalent norms induce the same topology.

5. If you had to make a choice, what would you say is the single most important thing you learned in class?

The Homework is due on Friday, March 14.