Let $\( V \)$ be a vector space over the complex numbers $\( \mathbb{C} \)$. A positive sesquilinear form on $\( V \)$ is a map $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} : V\times V \to \mathbb{C} \)$ (so that $\( \mathopen{}\left(x, y\right)\mathclose{}\mapsto \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \)$) that satisfies the following conditions for all $\( w \)$, $\( x \)$, and $\( y \)$ in $\( V \)$ and $\( λ \)$ in $\( \mathbb{C} \)$:

1. $\( \mathopen{}\left\langle{}x+λy, w\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, w\right\rangle\mathclose{}+λ\mathopen{}\left\langle{}y, w\right\rangle\mathclose{} \)$.
2. $\( \overline{ \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} }= \mathopen{}\left\langle{}y, x\right\rangle\mathclose{} \)$.
3. $\( \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}\geq 0 \)$.

Satisfying the first property in Definition I.1 is often refered to as being linear in the first slot. The first two properties together imply that a positive sesquilinear form is also conjugate linear in the second slot, meaning $\( \mathopen{}\left\langle{}w, x+λy\right\rangle\mathclose{}= \mathopen{}\left\langle{}w, x\right\rangle\mathclose{}+\overline{ λ }\mathopen{}\left\langle{}w, y\right\rangle\mathclose{} \)$.

Let $\( V \)$ be a vector space with positive sesquilinear form $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$. For all $\( x \)$ and $\( y \)$ in $\( V \)$, $\( {\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}\mathopen{}\left\langle{}y, y\right\rangle\mathclose{} \)$.

Let $\( V \)$ be a vector space with positive sesquilinear form $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$. For all $\( x \)$ and $\( y \)$ in $\( V \)$, $\( {\mathopen{}\left\langle{}x+y, x+y\right\rangle\mathclose{}}^{\frac{1}{2}}\leq {\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}}+{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}^{\frac{1}{2}} \)$.

An inner product $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$ on a vector space $\( V \)$ is a positive sesquilinear form such that $\( \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}= 0 \)$ if and only if $\( x= 0 \)$. A vector space equipped with an inner product is an inner product space.

Let $\( V \)$ equal $\( {\mathbb{C}}^{n} \)$. Given an $\( n \)$-by-$\( n \)$ matrix $\( M \)$ that is positive definite (meaning $\( M \)$ is invertible and $\( Mu\cdot u\geq 0 \)$ for all $\( u \)$ in $\( V \)$), define $\( \mathopen{}\left\langle{}u, v\right\rangle\mathclose{}= Mu\cdot v \)$ where $\( u\cdot v \)$ is the usual dot product of vectors $\( u \)$ and $\( v \)$ in $\( {\mathbb{C}}^{n} \)$.

Let $\( V \)$ be $\( \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}, \mathbb{C}\right)\mathclose{} \)$, the set of continuous complex-valued functions on the real interval $\( \mathopen{}\left[a, b\right]\mathclose{} \)$. For functions $\( φ \)$ and $\( ψ \)$ in $\( V \)$, define $\( \mathopen{}\left\langle{}φ, ψ\right\rangle\mathclose{} \)$ to be $\( \int _{ a}^{ b}{} φ\mathopen{}\left( t\right)\mathclose{}\overline{ ψ\mathopen{}\left( t\right)\mathclose{} } \,\mathrm{d}t \)$.

Let $\( V \)$ be the set of complex sequences that are eventually zero. For two such sequences $\( \mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty} \)$ and $\( \mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty} \)$, define $$\[ \mathopen{}\left\langle{}\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}, \mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty}\right\rangle\mathclose{}= \sum_{j=1}^{\infty}{} {α}_{j}\overline{{β}_{j}} \text{.} \]$$

Riffing on the previous examples, take $\( V \)$ as in Example I.8 but define the inner product by $$\[ \mathopen{}\left\langle{}\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}, \mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty}\right\rangle\mathclose{}= \int _{0}^{1}{} \mathopen{}\left(\sum_{n=1}^{\infty}{} {α}_{n}{t}^{n} \right)\mathclose{}\overline{ \mathopen{}\left(\sum_{m=1}^{\infty}{} {β}_{m}{t}^{m} \right)\mathclose{} } \,\mathrm{d}t= \sum_{m,n=1}^{\infty}{} \frac{{α}_{n}\overline{ {α}_{m} }}{m+n+1} \text{.} \]$$

A vector $\( x \)$ of a vector space $\( V \)$ is a null vector for a positive sesquilinear form $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$ on $\( V \)$ if $\( \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}= 0 \)$.

Let $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$ be a positive sesquilinear form on a vector space $\( V \)$. Prove that the set $\( N \)$ of null vectors is a subspace of $\( V \)$; that is, show that $\( \mathbb{C}N= N \)$ and $\( N+N= N \)$.

Let $\( S \)$ be a subspace of a vector space $\( V \)$. For elements $\( x \)$ and $\( y \)$ of $\( V \)$, write $\( x\equiv y \)$ if $\( x-y \)$ is in $\( S \)$. Prove that this defines an equivalence relation on $\( V \)$.

For a subspace $\( S \)$ of a vector space $\( V \)$, write $\( x+S \)$ for the equivalence class of the element $\( x \)$ under the equivalence relation of Exercise I.12, and write $\( V/S \)$ for the quotient space, which is the set of all equivalence classes.

Let $\( S \)$ be a subspace of a complex vector space $\( V \)$. Prove that $\( V/S \)$ is also a complex vector space.

Let $\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} \)$ be a positive sesquilinear form on a vector space $\( V \)$. Let $\( N \)$ be the subspace of null vectors. Then $\( \mathopen{}\left\langle{}x+N, y+N\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \)$ defines an inner product on $\( V/N \)$.

Prove Proposition I.15 as follows: Using Proposition I.3, show that $\( \mathopen{}\left\langle{}V, N\right\rangle\mathclose{}= 0= \mathopen{}\left\langle{}N, V\right\rangle\mathclose{} \)$. Next, use that $\( \mathopen{}\left\langle{}V, N\right\rangle\mathclose{}= 0= \mathopen{}\left\langle{}N, V\right\rangle\mathclose{} \)$ to show that the new inner product is well-defined. Finally, show that the properties of Definition I.5 are satisfied.

Let $\( G \)$ be a group. The complex group algebra $\( \mathbb{C}G \)$ is the complex vector space spanned by the set of symbols $\( \mathopen{}\left\{\, {δ}_{g}\,\middle\vert\, g\in G\,\right\}\mathclose{} \)$ with multiplication defined by $\( {δ}_{g}{δ}_{h}= {δ}_{gh} \)$.

Let $\( G \)$ be a group and suppose $\( φ : G \to \mathbb{C} \)$ is a function such that for each positive integer $\( k \)$, $$\[ \sum_{i,j=1}^{k}{} {λ}_{i}\overline{{λ}_{j}}φ\mathopen{}\left( {g}_{i}{g}_{j}^{-1}\right)\mathclose{} \geq 0 \]$$ for any complex numbers $\( {λ}_{1} \)$, …, $\( {λ}_{k} \)$ and group elements $\( {g}_{1} \)$, …, $\( {g}_{k} \)$ (such a function is called positive definite). You can then define a positive sesquilinear form on $\( \mathbb{C}G \)$ by $$\[ \mathopen{}\left\langle{}\sum_{g\in G}{}{κ}_{g}{δ}_{g}, \sum_{h\in G}{}{λ}_{h}{δ}_{h}\right\rangle\mathclose{}= \sum_{g,h}{} {κ}_{g}\overline{{λ}_{h}}φ\mathopen{}\left( g{h}^{-1}\right)\mathclose{} \text{.} \]$$ Letting $\( N \)$ be the subspace of null vectors, $\( \mathbb{C}G/N \)$ is a big deal in group representation theory.

A norm on a vector space $\( V \)$ is a function $\( \mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{} : V \to \mathbb{R} \)$ such that, for all $\( v\in V \)$ and $\( λ\in \mathbb{C} \)$,

1. $\( \mathopen{}\left\lVert{}v\right\rVert\mathclose{}\geq 0 \)$.
2. $\( \mathopen{}\left\lVert{}v\right\rVert\mathclose{}= 0 \)$ implies $\( v= 0 \)$.
3. $\( \mathopen{}\left\lVert{}λv\right\rVert\mathclose{}= \mathopen{}\left\lvert{}λ\right\rvert\mathclose{}\mathopen{}\left\lVert{}v\right\rVert\mathclose{} \)$.
4. $\( \mathopen{}\left\lVert{}v+w\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}v\right\rVert\mathclose{}+\mathopen{}\left\lVert{}w\right\rVert\mathclose{} \)$.

A vector space equipped with a norm is a normed linear space.

An inner product space with $\( \mathopen{}\left\lVert{}x\right\rVert\mathclose{}= {\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}} \)$ is a normed linear space.

Prove Proposition I.21.

The supremum norm (or infinity norm) of a function $\( f \)$ in $\( \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \)$ is given by $\( \mathopen{}\left\lVert{}f\right\rVert_\infty\mathclose{}= \max_{t\in \mathopen{}\left[a, b\right]\mathclose{}}{} \mathopen{}\left\lvert{}f\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{} \)$.

Prove that the supremum norm is a norm.

For a real number $\( p \)$ with $\( p\geq 1 \)$, the $\( p \)$-norm of a function $\( f \)$ in $\( \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \)$ is given by $\( \mathopen{}\left\lVert{}f\right\rVert_p\mathclose{}= {\mathopen{}\left(\int _{a}^{b}{}{\mathopen{}\left\lvert{}f\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}}^{p}\,\mathrm{d}t\right)\mathclose{}}^{\frac{1}{p}} \)$.

Prove that the $\( 1 \)$-norm is a norm.

Use Proposition I.21 and Example I.7 to prove that the $\( 2 \)$-norm is a norm.