Definition:Inner Product Space

Definition
Let $\Bbb F$ be a subfield of $\C$.

Let $V$ be a vector space over $\Bbb F$.

Let $\innerprod \cdot \cdot : V \times V \to \Bbb F$ be an inner product on $V$.

We say that $\struct {V, \innerprod \cdot \cdot}$ is an inner product space.

That is, an inner product space is a vector space together with an associated inner product.

Also known as
It is also sometimes known as a pre-Hilbert space because by the Completion Theorem we can extend an inner product space to its completion, so it becomes a Hilbert space.

Also see

 * Definition:Inner Product


 * Definition:Semi-Inner Product Space
 * Definition:Hilbert Space


 * Norm satisfying Parallelogram Law induced by Inner Product: Corollary gives a complete classification of the normed vector spaces over $\R$ that can be identified with an inner product space. (with the identification from the "Notes" section)