Is there any other inner product on $C[0,1]$ other than integral of product over [0,1]?

Question

Is there any other inner product on $C[0,1]$ other than integral of product over [0,1]?

Till now I just come across only one inner product on that space. I am interested in knowing other

Any help will be appreciated...

Thanks a lot

Answer 1

The first idea would be to have a weighted integral: $$ \langle f,g\rangle=\int_0^1f(x)g(x)w(x)dx, $$ where $w$ is a strictly positive function in $[0,1]$.

Another idea would be to have a countable dense set $\{x_1,x_2,\ldots\}$ in $[0,1]$, and then define, for example $$ \langle f,g\rangle= \sum_{k=1}^\infty2^{-k}f(x_k)g(x_k). $$

Answer 2

It sounds like you're talking about space over $\Bbb R$.

Note that for any (measurable) function $k:[0,1]\to (0,1]$, the function $$ \langle f, g \rangle_k = \int_0^1 k(x)f(x)g(x)\,dx $$ defines an inner product. There are inner products that don't fall into this category, such as those of the form $$ \langle f, g \rangle_{k,w} = \int_0^1 k(x)f(x)g(x)\,dx + \sum_{k} w_k f(w_k)g(w_k). $$ That being said, even these can be written in the form of $\langle f,g \rangle _k$ if we use a suitable distribution $k$.