Let $w(x)$ be a strictly positive continuous function on [a,b]. Define a form on $C[a,b]$ by the formula $\langle f,g \rangle _w = \int^b_a f(x)g(x)dx$ for $f,g \in C[a,b]$. Show that it is an inner product.
Ive never seen the inner product expressed as a integral before so I'm fairly stumped.
Any guidance would help,
Cheers
I am guessing that you meant $\langle f ,g \rangle_w = \int_a^b w(t)f(t)g(t) dt$.
You need (you are working in the reals here) to show symmetry (ie $\langle f ,g \rangle_w = \langle g, f \rangle_w $), linearity in the first argument (ie, $\langle \lambda f ,g \rangle_w = \lambda \langle f ,g \rangle_w $ and $\langle f+h ,g \rangle_w = \langle f ,g \rangle_w + \langle h ,g \rangle_w$) and positive definiteness (ie, $\langle f ,f \rangle_w \geq 0$, and $\langle f ,f \rangle_w = 0$ iff $f=0$).
The first two properties follow immediately from the commutativity of multiplication and linearity of the integral. The latter follows from properties of the integral when integrating non-negative functions.