TopologicalComplement 2007-09-15 15:21:23 2009-06-27 16:56:04 Definition complementary subspace topologically complementary topologically complemented % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsubsection{Definition} Let $X$ be a topological vector space and $M \subseteq X$ a \PMlinkname{closed}{ClosedSet} subspace. If there exists a closed subspace $N \subseteq X$ such that \begin{displaymath} M \oplus N = X \end{displaymath} we say that $M$ is {\bf topologically complemented}. In this case $N$ is said to be a {\bf topological complement} of $M$, and also $M$ and $N$ are said to be {\bf topologically complementary} subspaces. \subsubsection{Remarks} \begin{itemize} \item It is known that every subspace $M \subseteq X$ has an algebraic complement, i.e. there exists a subspace $N \subseteq X$ such that $M \oplus N = X$. The existence of topological complements, however, is not always assured. \item If $X$ is an Hilbert space, then each closed subspace $M \subseteq X$ is topologically complemented by its orthogonal complement $M^{\perp}$, i.e. \begin{displaymath} M \oplus M^{\perp} = X . \end{displaymath} \item Moreover, for Banach spaces the converse of the last paragraph also holds, i.e. if each closed subspace is topologically complemented then $X$ is isomorphic a Hilbert space. This is the \PMlinkname{Lindenstrauss-Tzafriri theorem}{CharacterizationOfAHilbertSpace}. \end{itemize}