
# 9.1 Inner Products

In this section, $$V$$ is a finite-dimensional, nonzero vector space over $$\mathbb{F}$$.

Definition 9.1.1.  An inner product on $$V$$ is a map
\begin{equation*}
\begin{split}
\inner{\cdot}{\cdot}:\;&V\times V \to \mathbb{F}\\
&(u,v) \mapsto \inner{u}{v}
\end{split}
\end{equation*}
with the following four properties.

1. Linearity in first slot: $$\inner{u+v}{w}=\inner{u}{w}+\inner{v}{w}$$ and $$\inner{au}{v}=a\inner{u}{v}$$ for all $$u,v,w\in V$$ and $$a\in \mathbb{F}$$;
2. Positivity: $$\inner{v}{v} \ge 0$$ for all $$v\in V$$;
3. Positive definiteness: $$\inner{v}{v}=0$$ if and only if $$v=0$$;
4. Conjugate symmetry: $$\inner{u}{v}=\overline{\inner{v}{u}}$$ for all $$u,v\in V$$.

Remark 9.1.2. Recall that every real number $$x\in\mathbb{R}$$ equals its complex conjugate. Hence, for real vector spaces, conjugate symmetry of an inner product becomes actual symmetry.

Definition 9.1.3.  An inner product space is a vector space over $$\mathbb{F}$$ together with an inner product $$\inner{\cdot}{\cdot}$$.

Example 9.1.4.  Let $$V=\mathbb{F}^n$$ and $$u=(u_1,\ldots,u_n), v=(v_1,\ldots,v_n)\in \mathbb{F}^n$$. Then we can define an inner product on $$V$$ by setting
\begin{equation*}
\inner{u}{v} = \sum_{i=1}^n u_i \overline{v}_i.
\end{equation*}
For $$\mathbb{F}=\mathbb{R}$$, this reduces to the usual dot product, i.e.,
\begin{equation*}
u\cdot v = u_1v_1+\cdots+u_n v_n.
\end{equation*}

Example 9.1.5.  Let $$V=\mathbb{F}[z]$$ be the space of polynomials with coefficients in $$\mathbb{F}$$.
Given $$f,g\in \mathbb{F}[z]$$, we can define their inner product to be
\begin{equation*}
\inner{f}{g} = \int_0^1 f(z)\overline{g(z)}dz,
\end{equation*}
where $$\overline{g(z)}$$ is the complex conjugate of the polynomial $$g(z)$$.

For a fixed vector $$w\in V$$, one can define a map $$T:V\to \mathbb{F}$$ by setting $$Tv=\inner{v}{w}$$. Note that $$T$$ is linear by Condition~1 of Definition~9.1.1. This implies, in particular, that $$\inner{0}{w}=0$$ for every $$w\in V$$. By conjugate symmetry, we also have $$\inner{w}{0}=0$$.

Lemma 9.1.6.  The inner product is anti-linear in the second slot, that is, $$\inner{u}{v+w}=\inner{u}{v}+\inner{u}{w}$$ and $$\inner{u}{av} =\overline{a}\inner{u}{v}$$ for all $$u,v,w\in V$$ and $$a\in \mathbb{F}$$.