# Inner Product Space/Hilbert Space Problem

1. Jun 17, 2010

### mattos90

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution
I solved question 3 without any difficulty, but I can't seem to make any progress on question 4.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 17, 2010

### 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