Inner Product

From MATH213

Used in Fourier Series analysis.

Theorem 1: Existence of Inner Product

If $f,g\in L^2([a,b])$ then $<f,g>$ exists and is finite.

Definition 2: Standard Inner product on L^2([a, b])

If $f$ and $g$ are complex valued functions in $L^2([a,b])$ then the standard inner product is $<f,g>=\frac{1}{b-a}{\int }_a^{b}f(t)\overline{g(t)}dt$

• See L2 Function

Is the Inner product the same as the Dot Product?

An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.

Related

• Dot Product