\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{F-norm}

\hypertarget{fnorms}{}\section*{{F-norms}}\label{fnorms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An F-norm is a non-homogeneous variant of a [[norm]]: a translation-invariant [[metric]] on a [[vector space]] that satisfies properties in between being a [[G-norm]] (on the underlying [[abelian group]] of the vector space) and being a norm. As with norms, there is a semi- variant.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $K$ be a [[topological field]] (typically the [[real numbers]] or the [[complex numbers]], but conceivably only a [[topological ring]], or at least a [[commutative ring|commutative]] one); we will call the elements of $K$ \emph{scalars}. Let $V$ be a [[vector space]] (or [[module over a ring|module]]) over $K$; we will call the elements of $V$ \emph{vectors}. Let $\|{-}\|$ be a [[function]] from (the underlying set of) $V$ to the set of [[real numbers]].

\begin{defn}
\label{Gseminorm}\hypertarget{Gseminorm}{}
If

\begin{enumerate}%
\item ${\|0_V\|} = 0$ (or even just ${\|0\|} \leq 0$),
\item ${\|{-x}\|} = {\|x\|}$ (or even just ${\|{-x}\|} \leq {\|x\|}$) for each vector $x$, and
\item ${\|x + y\|} \leq {\|x\|} + {\|y\|}$ for each vector $x$ and vector $y$ (the [[triangle inequality]]),

\end{enumerate}
then ${\|{-}\|}$ is a \textbf{[[G-seminorm]]}.

\end{defn}
This is enough to prove that ${\|x\|} \geq 0$ for each $x$ in $V$, making $(x,y) \mapsto {\|y - x\|}$ (precisely) a translation-invariant [[pseudometric]] on $V$.

Note that addition $(x,y) \mapsto x + y\colon V \times V \to V$ is a [[short map]] under this pseudometric and so certainly [[continuous map|continuous]].

\begin{defn}
\label{Fseminorm}\hypertarget{Fseminorm}{}
If

\begin{enumerate}%
\item $\|{-}\|$ is a G-seminorm and
\item scalar multiplication $(a,x) \mapsto a x\colon K \times V \to V$ is continuous (relative to the topology on $K$ and the pseudometric on $V$),

\end{enumerate}
then $\|{-}\|$ is an \textbf{F-seminorm}.

\end{defn}
If the topology on $K$ is given by an [[absolute value]] $|{-}|$, then we can go further:

\begin{defn}
\label{seminorm}\hypertarget{seminorm}{}
If

\begin{enumerate}%
\item ${\|x + y\|} \leq {\|x\|} + {\|y\|}$ for each vector $x$ and vector $y$ and
\item ${\|a x\|} = {|a|} {\|x\|}$ for each scalar $a$ and vector $x$,

\end{enumerate}
then $\|{-}\|$ is a \textbf{[[seminorm]]}.

\end{defn}
Every seminorm is automatically an F-seminorm.

No longer assuming anything further about $K$, there are some subsidiary definitions:

\begin{defn}
\label{Fnorm}\hypertarget{Fnorm}{}
If

\begin{enumerate}%
\item $\|{-}\|$ is an F-seminorm and
\item $x = 0_V$ whenever $x$ is a vector and ${\|x\|} = 0$,

\end{enumerate}
then $\|{-}\|$ is an \textbf{F-norm}.

\end{defn}
Thus an F-norm is precisely an F-seminorm whose induced pseudometric is a [[metric]]. (Compare the relationship between [[G-norms]] and [[norms]] with G-seminorms in \ref{Gseminorm} and seminorms in \ref{seminorm} above.)

\begin{defn}
\label{Fspace}\hypertarget{Fspace}{}
If

\begin{enumerate}%
\item $\|{-}\|$ is an F-norm and
\item $x$ converges under the metric on $V$ whenever $x$ is a [[net]] of vectors and $\lim_{i,j} {\|x_j - x_i\|} = 0$ in $\mathbb{R}$,

\end{enumerate}
then $(V,{\|{-}\|})$ is an \textbf{F-space}.

\end{defn}
In other words, an F-space is a vector space equipped with an F-norm whose induced metric is [[complete metric|complete]] (or equivalently such that the topology on $V$ is [[complete topological group|complete]]).

\begin{defn}
\label{Frechetspace}\hypertarget{Frechetspace}{}
If

\begin{enumerate}%
\item $(V,{\|{-}\|})$ is an F-space and
\item $(V,{\|{-}\|})$ is [[locally convex space|locally convex]] as a topological vector space,

\end{enumerate}
then $(V,{\|{-}\|})$ is a \textbf{[[Fréchet space]]}.

\end{defn}
Finally, $F Sp$ is the [[category]] whose [[objects]] are F-spaces and whose morphisms are [[short linear maps]]; that said, often people really study the [[essential image]] of that category within the category of [[topological vector spaces]], or [[equivalence of categories|equivalently]] the category whose objects are F-spaces and whose morphisms are [[continuous map|continuous]] linear maps. (This is especially so with Fr\'e{}chet spaces, which have a common alternative definition that makes no reference to a canonical metric.)

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

The usual examples of F-spaces that are not Fr\'e{}chet spaces are the [[Lebesgue spaces]] $l^p$ for $p \lt 0 \lt 1$. These use a modified $p$-[[p-norm|norm]] in which

\begin{displaymath}
{\|x\|_p} = \sum_i {|x_i|^p}
\end{displaymath}
(so without the $p$th root) to ensure that the [[triangle inequality]] (\ref{Gseminorm}.3) holds.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The uniqueness theorem for [[complete space|complete]] [[norms]] in [[dream mathematics]] applies also to F-norms: assuming [[excluded middle]], [[dependent choice]], and the (classically false) [[Borel property]], two complete F-norms on a given [[vector space]] over the [[real numbers]] must be [[topological equivalence|topologically equivalent]]. See [[norm\#dreamUnique]].

category: analysis

[[!redirects F-space]] [[!redirects F-spaces]] [[!redirects F space]] [[!redirects F spaces]] [[!redirects Fspace]] [[!redirects Fspaces]]

[[!redirects F-norm]] [[!redirects F-norms]] [[!redirects F norm]] [[!redirects F norms]] [[!redirects Fnorm]] [[!redirects Fnorms]]

[[!redirects F-seminorm]] [[!redirects F-seminorms]] [[!redirects F seminorm]] [[!redirects F seminorms]] [[!redirects Fseminorm]] [[!redirects Fseminorms]]



\end{document}