Let $C([a,b], \mathbb{R})$ be the space of continous functions $f:[a,b] \to \mathbb{R}$. I am looking for examples of inner products on this space. I know the inner product $\langle f,g \rangle = \int_a^b fg$ which can be generalized to $\langle f,g \rangle = \int_a^b fgh$ where $h \in C([a,b], \mathbb{R})$ is a non negative function. I haven't been able to find more examples for the space of continuous functions. I'd be especially interested in an inner product that doesn't depend on integration. Does anyone know more examples or a reference?
Thanks.
Best Answer
Let $(r_n)_{n \ge 1}$ be an enumeration of $[a,b]\cap \mathbb{Q}$ (or, more general, a dense sequence in $[a,b]$) and let $(c_n)_{n \ge 1}$ be any sequence in $(0, \infty)$ with $\sum_{n \ge 1} c_n < \infty$. Then, check that $$ \langle f,g \rangle := \sum_{n \ge 1} c_nf(r_n)g(r_n) $$ is an inner product on $C([a,b], \mathbb{R})$.