new vector spaces from old ones

This entry list methods that give new vector spacesMathworldPlanetmath from old ones.

  1. 1.

    Changing the field (complexificationPlanetmathPlanetmath, etc.)

  2. 2.
  3. 3.
  4. 4.

    direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of vectors spaces

  5. 5.
  6. 6.

    Tensor productPlanetmathPlanetmathPlanetmath of vector spaces (

  7. 7.

    The space of linear maps from one vector space to another, also denoted by Homk(V,W), or simply Hom(V,W), where V and W are vector spaces over the field k

  8. 8.

    The space of endomorphismsPlanetmathPlanetmath of a vector space. Using the notation above, this is the space Homk(V,V)=End(V)

  9. 9.

    dual vector space (, and bi-dual vector space. Using the notation above, this is the space Hom(V,k), or simply V*.

  10. 10.

    The annihilatorPlanetmathPlanetmathPlanetmathPlanetmath of a subspaceMathworldPlanetmath is a subspace of the dual vector space

  11. 11.

    Wedge productMathworldPlanetmathPlanetmath of vector spaces

  12. 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:VW is a linear map.

  1. 1.

    The kernel of L is a subspace of V.

  2. 2.

    The image of L is a subspace of W.

  3. 3.

    The cokernelMathworldPlanetmathPlanetmath of L is a quotient spaceMathworldPlanetmath of W.

Topological vector spaces

Suppose V is topological vector spaceMathworldPlanetmath.

  1. 1.

    If W is a subspace of V then its closurePlanetmathPlanetmath W¯ is also a subspace of V.

  2. 2.

    If V is a metric vector space then its completion V~ is also a (metric) vector space.

  3. 3.

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 dimensionMathworldPlanetmathPlanetmathPlanetmath correspond to certain subspaces of the tensor algebra of the vector space. These spaces include:

  1. 1.

    The space of Euclidean inner productsMathworldPlanetmath.

  2. 2.

    The space of Hermitian inner products.

  3. 3.

    the space of symplectic structures.

  4. 4.

    vector bundles

  5. 5.

    space of connections

Title new vector spaces from old ones
Canonical name NewVectorSpacesFromOldOnes
Date of creation 2013-03-22 15:31:08
Last modified on 2013-03-22 15:31:08
Owner matte (1858)
Last modified by matte (1858)
Numerical id 16
Author matte (1858)
Entry type Topic
Classification msc 16-00
Classification msc 13-00
Classification msc 20-00
Classification msc 15-00