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Problem 15. Let V be a vector space. The double dual of V, denoted by V", is defined to be the dual space of V'. In other words, V" (V')'. Define A: V → V" by (Av)(x) = 4(v) = for all v € V, € V'. 1. Show that A is a linear map. 2. Show that if T = L(V), then T" o A = AoT, where T" = (T')'. 3. Show that if V is finite dimensional, then A is an isomorphism from V onto V".

Problem 15. Let V be a vector space. The double dual of V, denoted by V", is defined to be the dual space of V'. In other words, V" (V')'. Define A: V → V" by (Av)(x) = 4(v) = for all v € V, € V'. 1. Show that A is a linear map. 2. Show that if T = L(V), then T" o A = AoT, where T" = (T')'. 3. Show that if V is finite dimensional, then A is an isomorphism from V onto V".

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 15. Let V be a vector space. The double dual of V, denoted by V", is defined to be the dual space of V'. In other words, V" = (V')'. Define A : V → V" by (Av) (y) = 4(v) for all v € V, y € V'. 1. Show that A is a linear map. 2. Show that if T = L(V), then T" o A = A o T, where T" = (T')'. 3. Show that if V is finite dimensional, then A is an isomorphism from V onto V". Remark: Suppose V is finite-dimensional. Then V and V' are isomorphic, but finding an isomorphism from V onto V' generally requires choosing a basis of V. In contrast, the isomorphism A from V onto V" does not require a choice of basis and thus is considered more natural.
Transcribed Image Text:Problem 15. Let V be a vector space. The double dual of V, denoted by V", is defined to be the dual space of V'. In other words, V" = (V')'. Define A : V → V" by (Av) (y) = 4(v) for all v € V, y € V'. 1. Show that A is a linear map. 2. Show that if T = L(V), then T" o A = A o T, where T" = (T')'. 3. Show that if V is finite dimensional, then A is an isomorphism from V onto V". Remark: Suppose V is finite-dimensional. Then V and V' are isomorphic, but finding an isomorphism from V onto V' generally requires choosing a basis of V. In contrast, the isomorphism A from V onto V" does not require a choice of basis and thus is considered more natural.
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