## Banach and Hilbert spaces

Let $ℂ$ be the field of complex numbers. A complex vector space is an abelian group $V$ with a function $ℂ×V\to V$ such that

(a)   If $c\in ℂ$ and ${v}_{1},{v}_{2}\in V$ then $c\left({v}_{1}+{v}_{2}\right)=c{v}_{1}+c{v}_{2}$,
(b)   If ${c}_{1},{c}_{2}\in ℂ$ and $v\in V$ then $\left({c}_{1}+{c}_{2}\right)v={c}_{1}v+{c}_{2}v$,
(c)   If ${c}_{1},{c}_{2}\in ℂ$ and $v\in V$ then ${c}_{1}\left({c}_{2}v\right)=\left({c}_{1}{c}_{2}\right)v$,
(d)   If $v\in V$ then $1v=v$.
Let $X$ and $Y$ be complex vector spaces. A linear transformation from $X$ to $Y$ is a function $T:X\to Y$ such that
 if ${c}_{1},{c}_{2}\in ℂ$ and ${x}_{1},{x}_{2}\in X$     then     $T\left({c}_{1}{x}_{1}+{c}_{2}{x}_{2}\right)={c}_{1}T\left({x}_{1}\right)+{c}_{2}T\left({x}_{2}\right)$.
The morphisms in the category of vector spaces are linear transformations.

A topological vector space is a complex vector space $V$ with a topology such that addition and scalar multiplication are continuous maps. The morphisms in the category of topological vector spaces are continuous linear transformations.

Let $V$ be a complex vector space. A set $C\subseteq V$ is convex if $C$ satisfies

 if $x,y\in C,t\in \left[0,1\right]$ then $tx+\left(1-t\right)y\in C$.
A topological vector space $V$ is locally convex if $V$ has a basis of neighbourhoods of 0 consisting of convex sets.

Let $V$ be a complex vector space and let $C\subseteq V$ be a convex subset of $V$. A function $f:C\to ℝ$ is convex if $f$ satisfies

 if $x,y\in C,t\in \left[0,1\right]$ then $f\left(tx+\left(1-t\right)y\right)\le tf\left(x\right)+\left(1-t\right)f\left(y\right)$.

HW: Show that the exponential function $\mathrm{exp}:\left(a,b\right)\to ℝ$ is convex.

A normed linear space is a complex vector space $V$ with a function $‖\phantom{\rule{0.2em}{0ex}}‖:V\to {ℝ}_{\ge 0}$ such that

(a)   if $x,y\in V$ then $‖x+y‖\le ‖x‖+‖y‖$,
(b)   if $c\in ℂ$ and $v\in V$ then $‖cv‖=|c|‖v‖$,
(c)   if $v\in V$ and $‖v‖=0$ then $v=0$.

HW: Show that if $V$ is a normed linear space then the map $‖\phantom{\rule{0.2em}{0ex}}‖:V\to {ℝ}_{\ge 0}$ is uniformly continuous, $V$ is a metric space with respect to the metric $d:V×V\to {ℝ}_{\ge 0}$ defined by

 $d\left(x,y\right)=‖x-y‖$
and, with the metric space topology, $V$ is a topological vector space.

A Banach space is a normed linear space $X$ such that $X$ is a complete metric space with respect to the metric $d:X×X\to {ℝ}_{\ge 0}$ defined by

 $d\left(x,y\right)=‖x-y‖$.
Let $X$ and $Y$ be normed linear spaces. An isometry from $X$ to $Y$ is a linear transformation $T:X\to Y$ such that
 if $x\in X$     then     $‖Tx‖=‖x‖$.
Let $X$ and $Y$ be normed linear spaces. The norm of a linear transformation $T:X\to Y$ is
 $‖T‖=\mathrm{sup}\left\{‖Tx‖\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x\in X\phantom{\rule{0.2em}{0ex}}\text{such that}\phantom{\rule{0.2em}{0ex}}‖x‖\le 1\right\}$.
A linear transformation is bounded if $‖T‖<\infty$.

HW: If $X$ and $Y$ are normed linear spaces such that points are closed then a linear transformation $T:X\to Y$ is continuous iff it is bounded (reference???)

HW: Show that if $X$ and $Y$ are normed linear spaces then $B\left(X,Y\right)=\left\{\text{bounded linear transformations}\phantom{\rule{0.5em}{0ex}}\phi :X\to Y\right\}$ with $‖\phantom{\rule{0.2em}{0ex}}‖$ is a normed linear space and that if $Y$ is a Banach space then $B\left(X,Y\right)$ is a Banach space.

A Hilbert space is a complex vector space $V$ with a function $⟨,⟩:V×V\to ℂ$ such that

(a)   if ${v}_{1},{v}_{2}\in V$ then $⟨{v}_{1},{v}_{2}⟩=\stackrel{‾}{⟨{v}_{2},{v}_{1}⟩}$,
(b)   if ${c}_{1},{c}_{2}\in ℂ$ and ${v}_{1},{v}_{2},{v}_{3}\in V$ then $⟨{c}_{1}{v}_{1}+{c}_{2}{v}_{2},{v}_{3}⟩={c}_{1}⟨{v}_{1},{v}_{3}⟩+{c}_{2}⟨{v}_{2},{v}_{3}⟩,$
(c)   if $v\in V$ and $⟨v,v⟩=0$ then $v=0$,
(d)   $V$ is a Banach space with norm $‖\phantom{\rule{0.2em}{0ex}}‖:V\to {ℝ}_{\ge 0}$  given by      ${‖v‖}^{2}=⟨v,v⟩$.

Let $V$ be a Hilbert space and let $T:V\to V$ be a linear transformation. The adjoint of $T$ is the linear transformation

 ${T}^{*}:V\to V$    defined by     $⟨T{v}_{1},{v}_{2}⟩=⟨{v}_{1},{T}^{*}{v}_{2}⟩$,
for ${v}_{1},{v}_{2}\in V$. The linear transformation $T:V\to V$ is unitary if $T$ satisfies
 if ${v}_{1},{v}_{2}\in V$     then     $⟨T{v}_{1},T{v}_{2}⟩=⟨{v}_{1},{v}_{2}⟩$.

## Duals

Let $X$ with $‖\phantom{\rule{0.2em}{0ex}}‖:V\to {ℝ}_{\ge 0}$ be a normed linear space. Define

 ${X}^{*}=\left\{\phi :X\to ℂ\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\phi \phantom{\rule{0.5em}{0ex}}\text{is a linear transformation and}\phantom{\rule{0.5em}{0ex}}‖\phi ‖<\infty \right\},$
where
 $‖\phi ‖=\mathrm{sup}\left\{‖\phi \left(x\right)‖\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}x\in X\phantom{\rule{0.2em}{0ex}}\text{such that}\phantom{\rule{0.2em}{0ex}}‖x‖\le 1\right\}$.
Then, see [Ru, 5.21 and Ch. 5 Ex. 8],
(a)   ${X}^{*}$ is a Banach space.
(b)   ${X}^{*}$ separates points on $X$, i.e. if ${x}_{1},{x}_{2}\in X$ and ${x}_{1}\ne {x}_{2}$ then there exists $\phi \in {X}^{*}$ such that $\phi \left({x}_{1}\right)\ne \phi \left({x}_{2}\right)$.
(c)   The map
 $\begin{array}{rccl}\iota :& X& ⟶& {X}^{**}\\ & \begin{array}{c}x\\ \\ \end{array}& \begin{array}{c}⟼\\ \\ \end{array}& \begin{array}{cccc}{\iota }_{x}:& {X}^{*}& \to & ℂ\\ & \phi & ↦& \phi \left(x\right)\end{array}\end{array}$
is an injective linear map such that $‖{\iota }_{x}‖=‖x‖$.
The construction of $f$ in part (b) is a special case (or corollary) of the Hahn-Banach theorem, see [Ru, 5.21 and Theorem 5.20].

If $M$ is a subspace of a normed linear space $X$ and $\phi :M\to ℂ$ is a bounded linear functional then there exists a bounded linear functional $\Phi :X\to ℂ$ such that

(a)   if $m\in M$ then $\Phi \left(m\right)=\phi \left(m\right)$,
(b)   $‖\Phi ‖=‖\phi ‖$.

The proof of this theorem is essentially by induction, where the induction step extends $\phi$ from $M$ to $M+ℂ{x}_{0}$ for a vector ${x}_{0}$ which is not in $M$.

## Notes and References

These notes were synthesized from [Ru], [Kirillov], ... ????? They have evolved over the years through graduate courses in "Representation Theory" at University of Wisconsin, Madison and a course in "Measure Theory" at the Masters level at University of Melbourne. This presentation follows [Ru, Chapters 4 and 5].

## References

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.