For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.
Given vectors $x = (x_1, x_2, \dotsc, x_n)$ and $y = (y_1, y_2, \dotsc, y_n)$ in $\mathbb{R}^n$, the dot product of $x$ and $y$ is $$ x \cdot y = \sum_{j=1}^{n} x_j y_j. $$ The dot product on $\mathbb{R}^n$ is linear in both $x$ and $y$ and has the property that $x\cdot x \ge 0$ for all $x$, with equality if and only if $x = 0$. Moreover $x \cdot y = \lVert x\rVert \lVert y\rVert \cos(\theta)$, where $\lVert x\rVert$ denotes the length of $x$ and $\theta$ is the measure of the angle between the vectors $x$ and $y$. The dot product is then an algebraic tool which can be used to describe geometric properties of $\mathbb{R}^n$ (e.g. distance and angle).
An inner product is a generalization of the dot product. An inner product space is a vector space over a field $\mathbb K$ (either $\mathbb R$ or $\mathbb C$) endowed with a map $\langle\cdot,\cdot\rangle\colon V\times V\longrightarrow\mathbb K$ such that
- $(\forall v_1,v_2,v\in V):\langle v_1+v_2,v\rangle=\langle v_1,v\rangle+\langle v_2,v\rangle$;
- $(\forall v_1,v_2\in V)(\forall\lambda\in\mathbb{K}):\langle\lambda v_1,v_2\rangle=\lambda\langle v_1,v_2\rangle$;
- $(\forall v_1,v_2\in V):\langle v_1,v_2\rangle=\overline{\langle v_2,v_1\rangle}$;
- $(\forall v\in V):\langle v,v\rangle\geqslant0$ and $\langle v,v\rangle=0\iff v=0$.
Such a map is called an inner product. As an example, consider the space $\mathcal{C}\bigl([0,1]\bigr)$ of all continuous functions from $[0,1]$ into $\mathbb C$. If $f,g\in\mathcal{C}\bigl([0,1]\bigr)$, define$$\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}\ \mathrm dt.$$
