Real Inner Product Norms

Why

To every real-valued inner product there corresponds a norm, in a similar manner to the construction for $\R ^n$.

Let $(V, \F )$ be a vector space. Let $f: V \times V \to \F $ be an inner product with $f(x, x) \in \R $. Let $g: V \to \R $ such that

\[ g(x) = \sqrt{f(x, x)}. \]

Then $g$ is a norm.

The norm of a vector in an inner product space is the square root of the inner product of the vector with itself.