Inner product

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In mathematics, an inner product is an abstract notion on general vector spaces that is a generalization of the concept of the dot product in the Euclidean spaces. Among other things, the inner product on a vector space makes it possible to define the geometric operation of projection onto a closed subspace (in the metric topology induced by the inner product), just like how the dot product makes it possible to define, in the Euclidean spaces, the projection of a vector onto the subspace spanned by a set of other vectors. The projection operation is a powerful geometric tool that makes the inner product a desirable convenience, especially for the purposes of optimization and approximation.

Formal definition of inner product

Let X be a vector space over a sub-field F of the complex numbers. An inner product $\langle \cdot,\cdot \rangle$ on X is a sesquilinear^[1] map from $X \times X$ to $\mathbb{C}$ with the following properties:

$\langle x,y\rangle=\overline{\langle y,x\rangle}\,\,\forall x,y \in X$
$\langle x,y\rangle=0\,\, \forall y \in X \Rightarrow x=0$
$\langle \alpha x_1+\beta x_2,y\rangle= \alpha \langle x_1, y\rangle+\beta \langle x_2, y\rangle$ $\forall \alpha,\beta \in F$ and $\forall x_1,x_2,y \in X$ (linearity in the first slot)
$\langle x,\alpha y_1+\beta y_2 \rangle= \bar\alpha \langle x, y_1\rangle +\bar\beta \langle x, y_2\rangle$ $\forall \alpha,\beta \in F$ and $\forall x,y_1,y_2 \in X$ (anti-linearity in the second slot)
$\langle x,x\rangle \geq 0\,\, \forall x \in X$ (in particular it means that $\langle x,x\rangle$ is always real)
$\langle x,x\rangle=0 \Rightarrow x=0$

Properties 1 and 2 imply that $\langle x,y\rangle=0\, \forall x \in X \Rightarrow y=0$ .

Note that some authors, especially those working in quantum mechanics, may define an inner product to be anti-linear in the first slot and linear in the second slot, this is just a matter of preference. Moreover, if F is a subfield of the real numbers $\mathbb{R}$ then the inner product becomes a bilinear map from $X \times X$ to $\mathbb{R}$ , that is, it becomes linear in both slots. In this case the inner product is said to be a real inner product (otherwise in general it is a complex inner product).

Norm and topology induced by an inner product

The inner product induces a norm $\|\cdot\|$ on X defined by $\|x\|=\langle x,x \rangle^{1/2}$ . Therefore it also induces a metric topology on X via the metric $d(x,y)=\|x-y\|$ .