Linearity and Orthogonality of Inner Product in Vector Space H

[squenshl]

Homework Statement

Denote the inner product of f,g $\in$ H by <f,g> $\in$ R where H is some(real-valued) vector space
a) Explain linearity of the inner product with respect to f,g. Define orthogonality.
b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be defined as (give reasons)
i) <f,g> = integral from 0 to 1 of f(x)g(x) dx?
ii) <f,g> = integral from 0 to 1 $\lambda$(x)f'(x)g'(x) dx? where prime denotes the derivative and $\lambda$(x) > 0 is a smooth function (assuming f',g' $\in$ H)?
iii) <f,g> = f(x)g(x)?

Homework Equations

The Attempt at a Solution

a) That is just trivial
b) Not too sure on any of them

 

[squenshl]

For i) since the integral is a real number (provided that f and g are continuous, differentiable etc.) once computed suggest that this will be an inner product, similarly for ii) and iii) (except the integral part for iii))

 

[HallsofIvy]

An inner product must be such that
$$<au+ bv, w>= a<u, w>+ b<v, w>$$
$$<u, v>= \overline{<v, u>}$$

Can you show that those are true for each of the given operations?

 

[squenshl]

Do I show positive definiteness <f,f> >= 0? and <f,f> = 0 if and only if f = 0

 

[squenshl]

I just proved all 3 axioms to be an inner product.
I got that all 3 operations are all defined as inner products, is this correct (I'm a little skeptical on ii))?
i) and iii) are easy.