I'm trying to solve this problem but the textbook has very airy explanations regarding how to do so. I think I can do the first part by just checking off that the 4 axioms of inner products hold.
I'm unsure what the second part is even asking.
Let $L^2((0,1))$ be the space of integrable functions such that $\int_0^1f(t)^2dt<\infty$. Show that $<f|g> = \int_0^1f(t)g(t)dt$ defines an inner product on $L^2((0,1))$.
Show that $B={sin(2{\pi}nt)}_{n\epsilon N}$ is a family of orthogonal vectors with respect to this scalar product.
Any help is appreciated. Thanks.