Inner Product of Functions

Intuition

The inner product of functions is exactly the regular dot product, just in infinite dimensions and with a different “weight”.

Since ℝ^𝑛 is discrete, each component has a weight of 1, whereas in function spaces each component has weight ”𝑑𝑥”.

On finite Euclidean space ℝ^𝑛:

• $(x_1,...,x_n)\cdot (y_1,...,y_n)=x_1\times y_1\times 1+\cdots +x_n\times y_n\times 1$

On sequence space:

• $(x_1,...)\cdot (y_1,...)=x_1\times y_1\times 1+\cdots$

On function space:

• $f\cdot g=f(a)\times g(a)\times dx+f(a+dx)g(a+dx)\times dx+\cdots ={\int }_a^{b}f(x)g(x)dx$

Resources

• Inner Products in Hilbert Space (of Functions)