Definition:Inner Product Space

Definition
An inner product space is a vector space together with an associated inner product.

Explanation
While not technically a normed vector space itself, an inner product space $\struct {V, \innerprod \cdot \cdot}$ is typically identified with the normed vector space $\struct {V, \norm \cdot}$, where $\norm \cdot$ is the inner product norm on $\struct {V, \innerprod \cdot \cdot}$.

Usually, no distinction is made between $\struct {V, \innerprod \cdot \cdot}$ and $\struct {V, \norm \cdot}$.

Indeed, in practice, definitions and theorems that only explicitly talk about normed vector spaces are freely applied to inner product spaces under this identification.

On, a distinction between inner product spaces and normed vector spaces is maintained for foundational definitions, such as those concerning bounded linear functionals, but on more advanced results such as the Hahn-Banach Theorem, the distinction is not seen as worth maintaining.

As the inner product norm induces a metric, we can apply theorems to $\struct {V, \innerprod \cdot \cdot}$ about metric spaces.

As a Normed Vector Space is Hausdorff Topological Vector Space, we can also apply theorems to $\struct {V, \innerprod \cdot \cdot}$ about Hausdorff topological vector spaces.

Also known as
Some texts refer to $\struct {V, \innerprod \cdot \cdot}$ as a scalar product space.

has a different definition for scalar product space.

Some texts refer to $\struct {V, \innerprod \cdot \cdot}$ as a pre-Hilbert space.

This is because by the Completion Theorem we can extend an inner product space to its completion, so it becomes a Hilbert space.

Also defined as
Some texts require that an inner product space is a vector space over $\R$ or $\C$.

This ensures that for all $v \in V$, the inner product norm:
 * $\norm v = \sqrt {\innerprod v v}$

is a scalar.

prefers the more general definition, and lists additional requirement on $\Bbb F$ in theorems where it is needed, such as the Gram-Schmidt Orthogonalization theorem.

Also see

 * Definition:Inner Product


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


 * Norm satisfying Parallelogram Law induced by Inner Product: Corollary