# Definition:Inner Product Space

## Definition

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

### Complex Inner Product Space

Let $V$ be a vector space over a complex subfield $\GF$.

Let $\innerprod \cdot \cdot : V \times V \to \GF$ be an complex inner product on $V$.

We say that $\struct {V, \innerprod \cdot \cdot}$ is a **(complex) inner product space**.

### Real Inner Product Space

Let $V$ be a vector space over a real subfield $\GF$.

Let $\innerprod \cdot \cdot : V \times V \to \GF$ be an real inner product on $V$.

We say that $\struct {V, \innerprod \cdot \cdot}$ is a **(real) inner product space**.

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$, 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 Topological Vector Space, we can also apply theorems to $\struct {V, \innerprod \cdot \cdot}$ about **topological vector spaces**.

## Also known as

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

$\mathsf{Pr} \infty \mathsf{fWiki}$ 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.

$\mathsf{Pr} \infty \mathsf{fWiki}$ 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

- Results about
**inner product spaces**can be found**here**.