\chapter{Spaces of functions and sequences}

Or: ``What I remember of functional analysis''.

There are many different function spaces in analysis (and generalized function spaces:
spaces of various generalizations of functions).
The way to make sense of them is to understand the context in which they are defined
(don't just read the definition).

why define?
what context?
- differentiation
  --> differentiable, C^k, distributions
- integration
  --> *lots* of spaces
- solving PDEs
  --> where these spaces came from

2 steps of abstraction in functional analysis:
- general theorems about topological vector spaces (and normed vector spaces)
- specific examples of topological vector spaces,
  which in turn arises from looking at particular question,
  like Fourier analysis or solving particular PDEs

Sobolev spaces
[BTW, there *is* a fractional derivative,
 but it is *not* local.]

lotta simple questions like:
"discontinuities of everywhere differentiable"
  -> Cantor and uncountable sets
     indeed, this is origin of Cantor set!
     (as discontinuities of certain differentiable function)
  ("descriptive set theory")

"convergence of Fourier series"
  -> lotta these wacky spaces

\section{Categories of function spaces}
Behavior at infinity:
\begin{itemize}
 \item compactly supported
 \item $C_0(X)$: vanishing at infinity
 \item bounded at infinity ($\limsup \abs{f}$ finite)
 \item $\cS(X)$: Scwartz space of rapidly decreasing functions
   (smooth functions such that they and all derivatives to all orders
    vanish at infinity faster than $x^{-n}$)
\end{itemize}

\begin{itemize}
 \item $L^p$
 \item $L^p_{\text{loc}}$
\end{itemize}
embeddings, duality

C^k
C^{k,\alpha}, esp. C^{1,\alpha}
D^k

Forms of continuity:
\begin{itemize}
 \item Lipschitz continuous
 \item Absolutely continuous
 \item Uniformly continuous
\end{itemize}


Sobolev spaces

AC
BV

various connections:
what's derivative?

- multiplication always defined
  (but what space does it land in?)
- convolution *not* always defined

Properly, you can define the product on the product space;
then multiplication is the pull-back under the diagonal,
and convolution is the push-forward under multiplication in the domain
(a ``co''diagonal; see more under bialgebras and Hopf algebras).


- differentiation always defined, as *distribution*

- duality
(hey, this is when the product is in $L^1$!
 this multiplication/convolution/duality
 is rather like the bialgebra/Hopf algebra world)

life much easier for sequences

\subsection{Convolution}


\section{Generalized functions}
Key generalizations of functions are:
\begin{itemize}
 \item measures
 \item distributions
\end{itemize}

\subsection{Measures}
Absolutely continuous means integral of a function


Lebesgue decomposition
- absolutely continuous
- singular

More finely:
- absolutely continuous
- singular (continuous)
- pure point

\subsection{Distributions}
Distributions can be motivated formally 

\section{Sequence spaces}
Recall the theme:
do everything for sequences, as discrete functions:
then no local / differential issues.

Sequences are exactly functions on $\bN$ (or dually, elements of the semigroup algebra
$R[\bN]$); bi-infinite sequences are exactly functions on $\bZ$ (or elements of the group algebra $R[\bZ]$).

Cauchy product = convolution

finite sequences = compactly supported functions


$c_0$ and $C_0$: null sequences/vanishing at infinity
$\ell^p$ and $L^p$: $p$-power integrable

convolution:
c_0 and everything
\ell^1 and \ell^\infty
