Is the Inner Product Space in C[0,2] Satisfied by the Integral Equality?

[Poirot1]

consider C[0,2], the set of continuous functions from [0,2] to C.

The inner product is <f,g> = the integral of f(t)g(t)* from 0 to 2. show that:

sqrt(2)||f|| is greater than or equal to the magnitude of the integral of f from 0 to 2, where ||.|| is the norm of f.

 

[ModusPonens]

What does g* mean?

 

[Poirot1]

conjugate

 

[Opalg]

consider C[0,2], the set of continuous functions from [0,2] to C.

The inner product is <f,g> = the integral of f(t)g(t)* from 0 to 2. show that:

sqrt(2)||f|| is greater than or equal to the magnitude of the integral of f from 0 to 2, where ||.|| is the norm of f.

What have you tried so far? Can you think of results that might help here (Cauchy–Schwarz inequality perhaps, for a suitable choice of g)?

 

[Poirot1]

What have you tried so far? Can you think of results that might help here (Cauchy–Schwarz inequality perhaps, for a suitable choice of g)?

I've tried writing what each side is. I don't see how schwarz inequality is relevant. I'm interested in f, not g.

 

[ModusPonens]

\(\displaystyle |\int_0^2 f(t)dt|=|2\bar{f}_1+2i\bar{f}_2|=\sqrt{4(\bar{f}_1)^2+4(\bar{f}_2)^2}=2\sqrt{(|\bar{f}|)^2}\leq 2\sqrt{\bar{|f|^2}}=2\sqrt{(\int_0^2 |f|^2 dt)/2}=\sqrt{2}||f||\)

where the bar is the average and \(\displaystyle f=f_1+if_2\).

EDIT: I made a correction and some clarifications of notation.