Math 2.0

Linear and multilinear algebra; matrix theory Linear maps Theorem

Canonical isomorphism with the double dual

Builds on Dual spaceDimension of the dual space

Statement

Let FF be a field and let VV be a vector space over FF. Let V∗∗=(V∗)∗V^{**} = (V^*)^* be the dual of the dual space. The evaluation map ev⁡ ⁣:V→V∗∗\operatorname{ev} \colon V \to V^{**} is defined by ev⁡(v)(f)=f(v)\operatorname{ev}(v)(f) = f(v) for v∈Vv \in V and f∈V∗f \in V^*. This map is linear and injective. If VV is finite-dimensional, then ev⁡\operatorname{ev} is an isomorphism.

Remarks

The evaluation map does not depend on a choice of basis, so it is called canonical or natural. In finite dimensions, it is an isomorphism because it is injective and dim⁡V∗∗=dim⁡V∗=dim⁡V\dim V^{**} = \dim V^* = \dim V. For infinite-dimensional VV, the evaluation map is not surjective.

Reading path

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear form, definition, no evidence levelLinear formdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionDual basis, definition, no evidence levelDual basisdefinitionDimension of the dual space, statement, no evidence levelDimension of the dual…statementCanonical isomorphism with the double dual, statement, no evidence levelCanonical isomorphism…statement
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  1. Vector space definition
  2. Linear combination definition
  3. Linear map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear form definition
  7. Linear span definition
  8. Space of linear maps definition
  9. Dual space definition
  10. Spanning set definition
  11. Basis definition
  12. Dual basis definition
  13. Dimension of the dual space statement
  14. Canonical isomorphism with the double dual statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_space, Injection into the double-dual
About this page
Tag0ZJM9D7NPS
URIm2://0ZJM9D7NPS
Kindstatement, theorem
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 0ZJM9D7NPS.
UsesDual space, Dimension of the dual space

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