# Normed linear space and inner product space

1. Oct 3, 2008

### cabin5

1. The problem statement, all variables and given/known data
Not all normed linear spaces are inner product spaces. Give examples.

2. Relevant equations
all equations and conditions constructing inner and normed linear spaces.

3. The attempt at a solution
Well, I tried some of spaces like L space, but I didn't find any logical solutions. Frankly, I don't know from where to begin to prove this.

2. Oct 3, 2008

### morphism

Try to flip through your book to find a property inner product spaces satisfy that is stated in terms of the norm generated by the inner product.

If you want another hint, post back.

3. Oct 3, 2008

### cabin5

I know all the sufficient properties for an inner product space, but how can I find a suitable example for this particular problem?

4. Oct 3, 2008

### morphism

Did you do what I suggested? What sufficient properties did you find?

5. Oct 3, 2008

### cabin5

ah, do you mean positive definiteness or parallelogram law?

6. Oct 3, 2008

### morphism

The latter is very useful!

7. Oct 3, 2008

### cabin5

yes,ok, but how should I write down the associated norm of a space which I don't know.

the only thing that I can write down is the parallelogram law itself which automatically yields the same thing in terms of inner product of its associated norm.

I am a bit confused...

8. Oct 3, 2008

### morphism

Work with a specific example. Take R^2 for instance. What norms do you know on this space?

9. Oct 3, 2008

### cabin5

thanks for the post!