new vector spaces from old ones
This entry list methods that give new vector spaces^{} from old ones.

1.
Changing the field (complexification^{}, etc.)
 2.
 3.

4.
direct product^{} of vectors spaces
 5.

6.
Tensor product^{} of vector spaces (http://planetmath.org/TensorProductClassical)

7.
The space of linear maps from one vector space to another, also denoted by ${\mathrm{Hom}}_{k}(V,W)$, or simply $\mathrm{Hom}(V,W)$, where $V$ and $W$ are vector spaces over the field $k$

8.
The space of endomorphisms^{} of a vector space. Using the notation above, this is the space ${\mathrm{Hom}}_{k}(V,V)=\mathrm{End}(V)$

9.
dual vector space (http://planetmath.org/DualSpace), and bidual vector space. Using the notation above, this is the space $\mathrm{Hom}(V,k)$, or simply ${V}^{*}$.

10.
The annihilator^{} of a subspace^{} is a subspace of the dual vector space

11.
Wedge product^{} of vector spaces

12.
A field $k$ is a vector space over itself. Consider a set $B$ and the set $V$ of all functions from $B$ to $k$. Then $V$ has a natural vector space structure. If $B$ is finite, then $V$ can be viewed as a vector space having $B$ as a basis.
Vector spaces involving a linear map
Suppose $L:V\to W$ is a linear map.

1.
The kernel of $L$ is a subspace of $V$.

2.
The image of $L$ is a subspace of $W$.

3.
The cokernel^{} of $L$ is a quotient space^{} of $W$.
Topological vector spaces
Suppose $V$ is topological vector space^{}.

1.
If $W$ is a subspace of $V$ then its closure^{} $\overline{W}$ is also a subspace of $V$.

2.
If $V$ is a metric vector space then its completion $\stackrel{~}{V}$ is also a (metric) vector space.

3.
The direct integral of Hilbert spaces provides a new Hilbert space^{}.
Spaces of structures and subspaces of the tensor algebra of a vector space
There are also certain spaces of interesting structures on a vector space that at least in the case of finite dimension^{} correspond to certain subspaces of the tensor algebra of the vector space. These spaces include:

1.
The space of Euclidean inner products^{}.

2.
The space of Hermitian inner products.

3.
the space of symplectic structures.

4.
vector bundles

5.
space of connections
Title  new vector spaces from old ones 

Canonical name  NewVectorSpacesFromOldOnes 
Date of creation  20130322 15:31:08 
Last modified on  20130322 15:31:08 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  16 
Author  matte (1858) 
Entry type  Topic 
Classification  msc 1600 
Classification  msc 1300 
Classification  msc 2000 
Classification  msc 1500 