How would one prove that the map $\langle \cdot, \cdot \rangle: (f,g) \mapsto \int_{a}^{b}f(t)g(t)dt$ is an inner product?

Question

How would one prove that the map $\langle \cdot, \cdot \rangle: (f,g) \mapsto \int_{a}^{b}f(t)g(t)dt$ is an inner product?

We were taught to use this function for inner product related questions, but now we are being asked to prove it and I don't know where to start.

Thank you!

Answer 1

This came from the intuitive meaning of inner product.
Let $\vec{a}$ and $\vec{b}$ are two vectors such that
$\vec{a}=a_1e_1+a_2e_2+.....\\ \vec{b}=b_1e_1+b_2e_2+...$ where $e_1,e_2,...$ are basis.
Then their inner product can be difined as
$$\left(a,b\right)=a_1b_1+a_2b_2+....\\ \Rightarrow\left(a,b\right)=\sum_{i=1}^{n}{a_ib_i}$$
and assume $f(x)=a_i$ when $x=i$ and $g(x)=b_i$ when $x=i$ , and also if $n\rightarrow\infty$ , then we can replace summation by integration,
$$\left(f,g\right)=\int{f(x)g(x)dx}$$