linear manifoldLinearManifold2003-11-26 14:00:032005-10-29 12:25:00Definitionhyperplane% this is the default PlanetMath preamble. as your knowledge
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Suppose $V$ is a vector space and suppose that $L$ is a
non-empty subset of $V$. If there exists a $v\in V$ such that $L+v=\{ v+l \mid l\in L\}$
is a vector subspace of $V$, then $L$ is a {\bf linear manifold} of $V$. Then we
say that the dimension of $L$ is the dimension of $L+v$ and write $\dim L = \dim (L+v)$.
In the important case $\dim L = \dim V -1$, $L$ is called a {\bf hyperplane}.
A linear manifold is, in other words, a linear subspace that has possibly been
shifted away from the origin.
For instance, in $\sR^2$ examples of linear
manifolds are points, lines (which are hyperplanes), and $\sR^2$ itself.
In $\sR^n$ hyperplanes naturally describe tangent planes to a smooth
hyper surface.
\begin{thebibliography}{9}
\bibitem{cristescu} R. Cristescu, \emph{Topological vector spaces},
Noordhoff International Publishing, 1977.
\end{thebibliography}