Definition:Inner Product

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Definition

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

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

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


An inner product is a mapping $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ that satisfies the following properties:

\((1)\)   $:$   Conjugate Symmetry      \(\displaystyle \forall x, y \in V:\) \(\displaystyle \quad \left \langle {x, y} \right \rangle = \overline{\left \langle {y, x} \right \rangle} \)             
\((2)\)   $:$   Bilinearity      \(\displaystyle \forall x, y \in V, \forall a \in \mathbb F:\) \(\displaystyle \quad \left \langle {a x + y, z} \right \rangle = a \left \langle {x, z} \right \rangle + \left \langle {y, z} \right \rangle \)             
\((3)\)   $:$   Non-Negative Definiteness      \(\displaystyle \forall x \in V:\) \(\displaystyle \quad \left \langle {x, x} \right \rangle \in \R_{\ge 0} \)             
\((4)\)   $:$   Positiveness      \(\displaystyle \forall x \in V:\) \(\displaystyle \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V \)             

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


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

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

\((1^\prime)\)   $:$   Symmetry      \(\displaystyle \forall x, y \in V:\) \(\displaystyle \left \langle {x, y} \right \rangle = \left \langle {y, x} \right \rangle \)             


Inner Product Space

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


Also known as

  • Innerproduct.


Notation

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


Also see