An inner product space is a vector space over $\mathbb K$ (that is, $\mathbb R$ or $\mathbb C$) endowed with a map $\langle\cdot,\cdot\rangle\colon V\times V\longrightarrow\mathbb K$ such that
1. $(\forall v_1,v_2,v\in V):\langle v_1+v_2,v\rangle=\langle v_1,v\rangle+\langle v_2,v\rangle$;
2. $(\forall v_1,v_2\in V)(\forall\lambda\in\mathbb{K}):\langle\lambda v_1,v_2\rangle=\lambda\langle v_1,v_2\rangle$;
3. $(\forall v_1,v_2\in V):\langle v_1,v_2\rangle=\overline{\langle v_2,v_1\rangle}$;
4. $(\forall v\in V):\langle v,v\rangle\geqslant0$ 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$$\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}\ \mathrm dt.$$