Complex Inner Product

Definition

The complex inner product (or dot product, scalar product) of two complex vectors $x, y \in \C ^n$ is

\[ x_1y_1 + x_2y_2 + \cdots + x_ny_n \]

We denote the inner product of $x$ and $y$ by $\ip{x,y}$.

An inner product space is tuple whose first object is a vector space over the real or complex numbers and whose second object is a conforming inner product.

Other terminlogy

Some older authors use the term pre-Hilbert space.