Definition:Inner Product

From ProofWiki
Jump to navigation Jump to search


Let $\C$ be the field of complex numbers.

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

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

An inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the following properties:

\((1)\)   $:$   Conjugate Symmetry      \(\ds \forall x, y \in V:\) \(\ds \quad \innerprod x y = \overline {\innerprod y x} \)             
\((2)\)   $:$   Linearity in first argument      \(\ds \forall x, y \in V, \forall a \in \GF:\) \(\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \)             
\((3)\)   $:$   Non-Negative Definiteness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x \in \R_{\ge 0} \)             
\((4)\)   $:$   Positiveness      \(\ds \forall x \in V:\) \(\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V \)             

That is, an inner product is a semi-inner product with the additional condition $(4)$.

If $\GF$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$.

Then $(1)$ above may be replaced by:

\((1')\)   $:$   Symmetry      \(\ds \forall x, y \in V:\) \(\ds \innerprod x y = \innerprod y x \)             

Inner Product Space

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

Also known as

  • Innerproduct

Also denoted as

$\innerprod x y$ is also denoted as $\left \langle {x; y} \right \rangle$.

If there is more than one vector space under consideration, then the notation $\innerprod x y_V$ for a vector space $V$ is commonplace.

Also see