# Definition:Semi-Inner Product

## Definition

### Complex Semi-Inner Product

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

A (complex) semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the (complex) semi-inner product axioms:

 $(1)$ $:$ Conjugate Symmetry $\ds \forall x, y \in V:$ $\ds \quad \innerprod x y = \overline {\innerprod y x}$ $(2)$ $:$ Sesquilinearity $\ds \forall x, y, z \in V, \forall a \in \GF:$ $\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z$ $(3)$ $:$ Non-Negative Definiteness $\ds \forall x \in V:$ $\ds \quad \innerprod x x \in \R_{\ge 0}$

### Real Semi-Inner Product

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

A (real) semi-inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the real semi-inner product axioms:

 $(1^\prime)$ $:$ Symmetry $\ds \forall x, y \in V:$ $\ds \innerprod x y = \innerprod y x$ $(2)$ $:$ Sesquilinearity $\ds \forall x, y, z \in V, \forall a \in \GF:$ $\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z$ $(3)$ $:$ Non-Negative Definiteness $\ds \forall x \in V:$ $\ds \quad \innerprod x x \in \R_{\ge 0}$

### Semi-Inner Product Space

A semi-inner product space is a vector space together with an associated semi-inner product.

## Examples

### Sequences with Finite Support

Let $\GF$ be a subfield of $\C$.

Let $V$ be the vector space of sequences with finite support over $\GF$.

Let $\innerprod \cdot \cdot: V \times V \to \GF$ be the mapping defined by:

$\ds \innerprod {\sequence {a_n} } {\sequence {b_n} } = \sum_{n \mathop = 1}^\infty a_{2 n} \overline {b_{2 n} }$

Then $\innerprod \cdot \cdot$ is a semi-inner product on $V$ but not an inner product on $V$.