Normed linear space and inner product space

[cabin5]

Homework Statement

Not all normed linear spaces are inner product spaces. Give examples.

Homework Equations

all equations and conditions constructing inner and normed linear spaces.

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.

 

[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.

 

[cabin5]

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

 

[morphism]

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

 

[cabin5]

ah, do you mean positive definiteness or parallelogram law?

 

[morphism]

The latter is very useful!

 

[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...

 

[morphism]

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

 

[cabin5]

thanks for the post!