Definition 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 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 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.

References

1

R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.