| Version current |
Version 1 |
| \subsubsection{Definition} |
\subsubsection{Definition} |
| Let $X$ be a topological vector space and $M \subseteq X$ a \PMlinkname{closed}{ClosedSet} subspace. |
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 |
If there exists a closed subspace $N \subseteq X$ such that |
| \begin{displaymath} |
\begin{displaymath} |
| M \oplus N = X |
M \oplus N = X |
| \end{displaymath} |
\end{displaymath} |
| we say that $M$ is {\bf topologically complemented}. |
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. |
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} |
\subsubsection{Remarks} |
| \begin{itemize} |
\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 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. |
\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} |
\begin{displaymath} |
| M \oplus M^{\perp} = X . |
M \oplus M^{\perp} = X . |
| \end{displaymath} |
\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}.
|
\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 a Hilbert space. This is the \PMlinkname{Lindenstrauss-Tzafriri theorem}{CharacterizationOfAHilbertSpace}.
|
| \end{itemize} |
\end{itemize} |
