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Let V be a nonzero vector space over a field F, and suppose that S is a basis for V. (By

Chapter 2, Problem 25

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QUESTION:

Let V be a nonzero vector space over a field F, and suppose that S is a basis for V. (By the corollary to Theorem 1.13 (p. 60) in Section 1.7, every vector space has a basis). Let C(S, F) denote the vector space of all functions / !F(S, F) such that f(s) 0 for all but a finite number 110 Chap. 2 Linear Transformations and Matrices of vectors in 5. (See Exercise 14 of Section 1.3.) Let # : C{S,F) -> V be defined by *(/) = 0 if / is the zero function, and *(/) = E /* . sS,/(s)^0 otherwise. Prove that \I> is an isomorphism. Thus every nonzero vector space can be viewed as a space of functions.

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QUESTION:

Let V be a nonzero vector space over a field F, and suppose that S is a basis for V. (By the corollary to Theorem 1.13 (p. 60) in Section 1.7, every vector space has a basis). Let C(S, F) denote the vector space of all functions / !F(S, F) such that f(s) 0 for all but a finite number 110 Chap. 2 Linear Transformations and Matrices of vectors in 5. (See Exercise 14 of Section 1.3.) Let # : C{S,F) -> V be defined by *(/) = 0 if / is the zero function, and *(/) = E /* . sS,/(s)^0 otherwise. Prove that \I> is an isomorphism. Thus every nonzero vector space can be viewed as a space of functions.

ANSWER:

Step 1 of 4

Let V be a vector space and suppose that S is a basis for V. Let   denote the vector space of all functions  such that  for all but a finite number of vectors in S.

Let be the function defined by

         

To prove that is an isomorphism.

First, let us show that is linear.

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