Suppose we consider $C([a,b])$ and define an inner product on the vector space by $$\left<f,g\right>=\int_a^b f(t)g(t)dt$$ where $f,g$ are real valued functions on $[a,b]$.
My question is, can we visualize this inner product, i.e. how it graphically looks for two given continuous functions? I tried to visualize it as area under $f(t)g(t)$ curve between the two bounds but that does not help me much.I want a more clear visualization. A diagram would also help.
There may be better and more comprehensive ways to visualize this inner product, but it is rather hard given that $C[a,b]$ as a vector space over $\mathbb{R}$ is infinite dimensional. In general, if you know the type of functions you want to compare, you can consider one-parameter families of such functions and check their inner product as the parameters vary. Here are some examples of plots showing the inner product of one parameter families of functions. The second plot I scaled by a factor of $2$.
This by no means gives a comprehensive visualization of the inner product, since it only considers one parameter families, but it is one way I see a visual application.