What is the difference between product of two functions and dot product of functions?

Question

Given two functions e.g.: $$f(x)=\sin(x) \text{, } g(x)=2\cos(x)$$

Why can't we write: $$f\cdot g=\sin(2x)$$

But we use this: $$(f,g)=\int_{\alpha}^{\beta}f(x)g(x)dx$$ I don't quite understand if those two things are completely different and I'm mixing two totally different concepts.

Answer 1

First operation maps two functions to a function, namely their product.

Second one maps two functions to a number.

So indeed very different operations.