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9.1: Inner Products

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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 $$a\in \mathbb{F}$$.

Proof. For additivity, note that
\begin{equation*}
\begin{split}
\inner{u}{v+w} & = \overline{\inner{v+w}{u}} = \overline{\inner{v}{u}+\inner{w}{u}}\\
& = \overline{\inner{v}{u}} + \overline{\inner{w}{u}} = \inner{u}{v} + \inner{u}{w}.
\end{split}
\end{equation*}
Similarly, for anti-homogeneity, note that
\begin{equation*}
\inner{u}{av} = \overline{\inner{av}{u}} = \overline{a\inner{v}{u}}
= \overline{a} \overline{\inner{v}{u}} = \overline{a} \inner{u}{v}.
\end{equation*}

We close this section by noting that the convention in physics is often the exact opposite of what we have defined above. In other words, an inner product in physics is traditionally linear in the second slot and anti-linear in the first slot.