For given functions $f$ and $g$ in a Hilbert space $L^2(a,b)$, prove if the Pythagorean theorem is true for $f$ and $g$, then it is also true for $cf$ and $cg$, where $c$ is a constant. What would $<cf, cg>$ be?
Assume the Pythagorean Theorem holds for functions $f$ and $g$ in a Hilbert space $L^2(a,b)$. Then $||f||^2 + ||g||^2 = ||f+g||^2$.
I have the beginning of the proof above however I am having trouble continuing.
The assumption is that $$\|f\|^2+\|g\|^2=\|f+g\|^2.$$ Now, since $\|cf\|=|c|\,\|f\|$ for any scalar $c$ and vector $f$, $$ \|cf\|^2+\|cg\|^2=|c|\,\|f\|^2+|c|\,\|g\|^2=|c|\,(\|f\|^2+\|g\|^2)=|c|\,\|f+g\|^2 =\|cf+cg\|^2. $$