Inner Product

Let $V$ be a vector space over $\mathbb{R}$ or over the complex numbers $\mathbb{C}$. An inner product on $V$ is a function $\langle\cdot,\cdot\rangle:V\times V\to \mathbb{R}$ (real case) or $\langle\cdot,\cdot\rangle:V\times V\to \mathbb{C}$ (complex case) such that for all $u,v,w\in V$ and all scalars $a,b$:

1. Linearity in the first argument: $\langle au+bv,w\rangle=a\langle u,w\rangle+b\langle v,w\rangle$.
2. Conjugate symmetry: $\langle v,u\rangle=\overline{\langle u,v\rangle}$, where the bar denotes complex conjugation (in the real case this reduces to symmetry).
3. Positive-definiteness: $\langle v,v\rangle\ge 0$ and $\langle v,v\rangle=0$ if and only if $v=0$.

An inner product induces a norm by $\|v\|=\sqrt{\langle v,v\rangle}$ (compare the Euclidean norm as the standard example), and it defines orthogonality via the condition $\langle u,v\rangle=0$.

Examples:

• On Euclidean space $\mathbb{R}^k$, the standard inner product is $\langle x,y\rangle=\sum_{i=1}^k x_i y_i$.
• On $\mathbb{C}^k$, the standard (Hermitian) inner product is $\langle x,y\rangle=\sum_{i=1}^k x_i\,\overline{y_i}$.
• If $A$ is a symmetric positive definite $k\times k$ real matrix, then $\langle x,y\rangle=x^\top A y$ defines an inner product on $\mathbb{R}^k$.