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\begin{document}

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\section*{locally convex topological vector space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ContinuousLinearFunctionals}{Continuous linear functionals}\dotfill \pageref*{ContinuousLinearFunctionals} \linebreak
\noindent\hyperlink{coprobes_and_curves}{Co-Probes and curves}\dotfill \pageref*{coprobes_and_curves} \linebreak
\noindent\hyperlink{diagram_of_properties}{Diagram of properties}\dotfill \pageref*{diagram_of_properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A [[topological vector space]] is \textbf{locally convex} if it has a [[base of its topology]] consisting of [[convex subsets|convex]] [[open subsets]]. Equivalently, it is a vector space equipped with a [[gauge space|gauge]] consisting of [[seminormed vector space|seminorms]]. As with other topological vector spaces, a locally convex space (LCS or LCTVS) is often assumed to be [[Hausdorff space|Hausdorff]].

Locally convex (topological vector) spaces are the standard setup for much of contemporary [[functional analysis]].

A natural notion of [[smooth map]] between lctvs is given by [[Michal-Bastiani smooth maps]].

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

The category $lctvs$ is a [[symmetric monoidal category]] with the [[inductive tensor product]] and even a symmetric [[closed monoidal category]], where the internal homs are given by the space of continuous linear maps with the topology of pointwise convergence.

\hypertarget{ContinuousLinearFunctionals}{}\subsubsection*{{Continuous linear functionals}}\label{ContinuousLinearFunctionals}

One reason why locally convex [[topological vector spaces]] are important is that lots of [[continuous linear functionals]] exist on them, at least if one assumes an appropriate choice principle, e.g., [[axiom of choice]] or [[ultrafilter theorem]] (or just [[dependent choice]] for a [[separable space]]). This fact is encapsulated in the [[Hahn-Banach theorem]]; a nice exposition is given in \hyperlink{Tao09}{Tao 09}. By way of contrast, a [[topological vector space]] which is \emph{not} locally convex, such as the [[Lebesgue space]] $L^p([0, 1])$ where $0 \lt p \lt 1$, need not have any (nonzero) [[continuous linear functionals]] at all.

\begin{defn}
\label{DirectedSystemOfSeminorms}\hypertarget{DirectedSystemOfSeminorms}{}
\textbf{(directed system of seminorms)}

A family $\{p_q\}_{q \in Q}$ of [[seminorms]] on some [[real vector space]] $V$ is called \emph{directed} if

\begin{displaymath}
\underset{q_1, q_2 \in Q}{\forall}
  \left(
    \underset{q \in Q}{\exists}
    \left(
      \underset{C \in (0,\infty)}{\exists}
      \left(
        \underset{v \in V}{\forall}
        \left(
          \,
          C p_q(v) \leq max\{ p_{q_1}(v), p_{q_2}(v) \}
          \,
        \right)
      \right)
    \right)
  \right)
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{Infusino17}{Infusino 17} \href{http://www.math.uni-konstanz.de/~infusino/TVS-SS17/Lect10.pdf}{def. 4.2.15})

\begin{prop}
\label{MaximumEnvelopeOfSeminorms}\hypertarget{MaximumEnvelopeOfSeminorms}{}
\textbf{(maximum envelope of seminorms)}

Let $V$ be a [[real vector space]] equipped with a set $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$ of [[seminorms]].

Then the [[maxima]] of [[inhabited|non-empty]] [[finite set|finite]] [[subsets]] $J \subset Q$ of these seminorms

\begin{displaymath}
v \mapsto \left( \underset{q \in J}{max} p_q(v) \right)
\end{displaymath}
are themselves seminorms, and the set of them

\begin{displaymath}
\left\{
    \underset{q \subset J}{max} p_q(-)
  \right\}_{ { J \subset Q } \atop {\text{finite, non-empty}} }
\end{displaymath}
generate the same [[topological vector space|topology]] on $V$ as the original $\{p_q\}$ do. Moreover, the system of maximum seminorms evidently form a filtered system according to def. \ref{DirectedSystemOfSeminorms}.

\end{prop}
(e.g. \hyperlink{Infusino17}{Infusino 17} \href{http://www.math.uni-konstanz.de/~infusino/TVS-SS17/Lect10.pdf}{prop. 4.2.13, 4.2.14})

\begin{prop}
\label{AlternativeCharacterizationOfContinuityForLinearFunctionals}\hypertarget{AlternativeCharacterizationOfContinuityForLinearFunctionals}{}
\textbf{(characterization of continuity for linear functionals by norm-bounds)}

Let $V$ be a [[real vector space]] and $\tau$ a [[topological space|topology]] on $V$ that makes it a locally convex topological vector space, induced from a set of [[seminorms]] $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$. Consider a [[linear function]]

\begin{displaymath}
L \colon V \to \mathbb{R}
\end{displaymath}
\begin{enumerate}%
\item (directed system of seminorms) If the system of seminorms $\{p_q\}_{q \in Q}$ is directed (def. \ref{DirectedSystemOfSeminorms}) then $L$ is a [[continuous function]] with respect to $\tau$, hence is a [[continuous linear functional]], precisely if it is $q$-continuous for some $q \in Q$:

\begin{displaymath}
\left(
    L \,\text{continuous with respect to}\, \tau
  \right)
  \;\Leftrightarrow\;
  \underset{q \in Q}{\exists}
  \left(
    \underset{C \in (0,\infty)}{\exists}
    \left(
      \underset{v \in V}{\forall}
      \left(
    \,
    {\vert L(v)\vert}
    \leq 
    C p_q(v)
    \,
      \right)
    \right)
  \right)
  \,.
\end{displaymath}

\item (general system of seminorms) Together with prop. \ref{MaximumEnvelopeOfSeminorms} this means that for $\{p_q\}_{q \in Q}$ any set of seminorms (not necessarily directed), then $L$ is continuous precisely if there exists a [[inhabited set|inhabited]] [[finite set|finite]] [[subset]] of seminorms such that $L$ is bounded with respect to the [[maximum]] over this set:

\begin{displaymath}
\left(
    L \,\text{continuous with respect to}\, \tau
  \right)
  \;\Leftrightarrow\;
  \underset{q_1, \cdots, q_n \in Q}{\exists}
  \left(
    \underset{C \in (0,\infty)}{\exists}
    \left(
      \underset{v \in V}{\forall}
      \left(
    \,
    {\vert L(v)\vert}
    \leq 
    C \underset{k = 1, \cdots, n}{max} p_{q_n}(v)
    \,
      \right)
    \right)
  \right)
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
(e.g. \hyperlink{Infusino17}{Infusino 17}, \href{http://www.math.uni-konstanz.de/~infusino/Lect12.pdf}{prop. 4.6.1, corollary 4.6.2}), or remark 3-4 4. here: \href{https://www.math.ksu.edu/~nagy/func-an-2007-2008/lcvs-5.pdf}{pdf}

\hypertarget{coprobes_and_curves}{}\subsubsection*{{Co-Probes and curves}}\label{coprobes_and_curves}

The collections of [[continuous linear functionals]] on a LCTVS is used in a way analogous to the collection of [[coordinate]] [[projections]] $pr_i:\mathbb{R}^n\to \mathbb{R}$ out of a [[Cartesian space]]. For example, [[curves]] in a LCTVS over the reals can be composed with functionals to arrive at a collection of functions $\mathbb{R} \to \mathbb{R}$ which are analogous to the `components' of the curve.

In one respect, a locally convex TVS is a [[nice topological space]] in that there are enough co-probes by maps to the base field.

\hypertarget{diagram_of_properties}{}\subsubsection*{{Diagram of properties}}\label{diagram_of_properties}

[[!include diagram of LCTVS properties]]

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

\begin{itemize}%
\item [[Fréchet spaces]] are locally convex topological vector spaces

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item J. L. Taylor, \emph{Notes on locally convex topological vector spaces} (1995) (\href{http://www.math.utah.edu/~taylor/LCS.pdf}{pdf})


\item [[Terence Tao]], \emph{\href{http://terrytao.wordpress.com/2009/01/26/245b-notes-6-duality-and-the-hahn-banach-theorem/}{Duality and the Hahn-Banach theorem}}, 2009


\item Maria Infusino, \emph{\href{http://www.math.uni-konstanz.de/~infusino/TVS-SS17/teachingSS2017.html}{Topological vector spaces}} 2017



\end{itemize}
category: analysis

[[!redirects locally convex space]]

[[!redirects locally convex]] [[!redirects locally convex spaces]] [[!redirects locally convex vector space]] [[!redirects locally convex vector spaces]] [[!redirects locally convex topological vector space]] [[!redirects locally convex topological vector spaces]] [[!redirects locally convex TVS]] [[!redirects locally convex TVSs]] [[!redirects locally convex TVSes]] [[!redirects LCS]] [[!redirects LCSs]] [[!redirects LCSes]] [[!redirects LCTVS]] [[!redirects LCTVSs]] [[!redirects LCTVSes]]

[[!redirects locally convex Hausdorff space]] [[!redirects locally convex Hausdorff spaces]] [[!redirects Hausdorff locally convex space]] [[!redirects Hausdorff locally convex spaces]] [[!redirects hausdorff locally convex space]]



\end{document}