# Definitions > [!NOTE] Definition ((Complex) Inner Product) > Let $\mathbb{F}$ be a complex subfield and $V$ be a [[vector space]] over $\mathbb{F}.$ An inner product is a [[Function|mapping]] $\langle \cdot , \cdot \rangle : V\times V\to \mathbb{F}$ such that for all vectors $u,v,w\in V,$ and scalars $a\in \mathbb{F},$ we get: > 1. Conjugate symmetry: $\langle u,v\rangle = \overline{\langle u,v\rangle}$where for all $z\in \mathbb{C},$ $\overline{z}$ denotes the [[Complex Conjugate|complex conjugate]] of $z.$ > 2. Linearity in first argument: $\langle au+v,w\rangle=a\langle u,w\rangle+\langle v,w\rangle$ > 3. Positive-definitness: $\langle u,u\rangle\geq 0$with equality iff $u=0_{V}.$ > > $(V,\langle \cdot, \cdot \rangle)$ is known as an [[Inner Product Space]]. > [!NOTE] Definition (Real Inner Product) > Let $V$ be a vector space over a real subfield $\mathbb{F}.$ An inner product is a mapping $\langle \cdot , \cdot \rangle : V\times V\to \mathbb{F}$ that satisfies the following: > 1. Commutativity: $\langle v,w\rangle = \langle w,v\rangle$ for any $v,w\in V.$ > 2. Bilinearity: $\langle (\lambda_{1}v_{1}+\lambda_{2}v_{2}),w\rangle = \lambda_{1} \langle v_{1},w\rangle+ \lambda_{2}\langle v_{2},w\rangle$ for any $v_{1},v_{2},w\in V$ and $\lambda_{1}, \lambda_{2} \in \mathbb{F}.$ > 3. Positive Definite: For any $v\in V,$ $\langle v,v\rangle \geq 0,$ and furthermore $\langle v,v\rangle=0$ iff $v=0_{V}.$ # Properties Note that [[Inner Product is Sesquilinear|inner product is sesquilinear]], that is, ... The [[Inner Product is At Most Product of Vector Norms (Cauchy-Schwartz Inequality)|Cauchy-Schwartz inequality]] asserts that $\langle v,w\rangle^{2}\leq \langle v,v\rangle\langle w,w\rangle.$ By [[Inner Product induces Norm]], ... # Applications