What does $\langle f,g \rangle:=\frac{2}{T}\int f(x)g(x)dx, T>0$ mean?

Question

Simple question. In my analysis lecture we were discussing Fourier Approximation (real & complex) and I came across this: $$\langle f,g\rangle:=\frac{2}{T}\int f(x)g(x)dx, T>0$$ What does this mean? And more specifically what does $\langle f,g\rangle$ mean and what is the area of mathematics that it deals in?

Answer 1

Without the factor $2/T$, this would be called the $L^2$ inner product on the interval $[0,T]$. It is kind of like a generalization of the dot product, since you multiply each pair of function values and integrate (compared to multiplying each pair of vector entries and summing). As a consequence the function space $L^2$ shares some similarities with Euclidean space (and also numerous major differences).

Note that the factor $2/T$ is not really necessary, but including it ensures that $\sin(n \pi x/T)$ and $\cos(n \pi x/T)$ (for $n=1,2,\dots$) have norm $1$, which is convenient later.

In general $\langle x,y \rangle$ (drawn with the symbols "\langle" and "\rangle" in LaTeX) is notation for the inner product between $x$ and $y$ (in some space which is usually assumed obvious from context).

Answer 2

The answer above is really good, I would only add that the area of mathematics that deals with it is functional analysis.
I just found this notes, maybe they could be useful or interesting for you. \http://www1.maths.leeds.ac.uk/~kisilv/courses/math3263m.pdf