Inner product

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 (in the metric topology induced by the inner product) subspace, 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 map from $X\times X$ to $\mathbb {C}$ with the following properties:

1. $\langle x,y\rangle ={\overline {\langle y,x\rangle }}\,\,\forall x,y\in X$
2. $\langle x,y\rangle =0\,\,\forall y\in X\Rightarrow x=0$
3. $\langle \alpha x_{1}+\beta x_{2},y\rangle =\alpha \langle x_{1},y\rangle +\beta \langle x_{2},y\rangle$ $\forall \alpha ,\beta \in F$ and $\forall x_{1},x_{2},y\in X$ (linearity in the first slot)
4. $\langle x,\alpha y_{1}+\beta y_{2}\rangle ={\bar {\alpha }}\langle x,y_{1}\rangle +{\bar {\beta }}\langle x,y_{2}\rangle$ $\forall \alpha ,\beta \in F$ and $\forall x,y_{1},y_{2}\in X$ (anti-linearity in the second slot)
5. $\langle x,x\rangle \geq 0\,\,\forall x\in X$ (in particular it means that $\langle x,x\rangle$ is always real)
6. $\langle x,x\rangle =0\Rightarrow x=0$

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 $d(x,y)=\|x-y\|$.