Denote the inner product of f,g [itex]\in[/itex] H by <f,g> [itex]\in[/itex] 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 [itex]\lambda[/itex](x)f'(x)g'(x) dx? where prime denotes the derivative and [itex]\lambda[/itex](x) > 0 is a smooth function (assuming f',g' [itex]\in[/itex] 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