Inner Product Space/Hilbert Space Problem

[mattos90]

Homework Statement

3. If z is any fixed element of an inner product space X, show that f(x) = <x,z> defines a bounded linear functional f on X, of norm ||z||.
4. Consider Prob. 3. If the mapping X --> X' (the space of continuous linear functionals) given by z |--> f is surjective, show that X must be a Hilbert space.

Homework Equations

The Attempt at a Solution

I solved question 3 without any difficulty, but I can't seem to make any progress on question 4.

 

[Coto]

To start you off, write down explicitly what it means for a map to be surjective and write down the requirements for something to be a Hilbert space.

What are your ideas about showing that the map z |--> f is surjective?

How would you show each of the requirements for a Hilbert space?

Coto