# Inner product

## 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)?

## The Attempt at a Solution

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

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
Homework Helper
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?

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

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.