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Math Help - Need a proof for a subspace

  1. #1
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    Need a proof for a subspace

    Hi all,

    The question goes: Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.

    Does this involve utilizing the fundamental theorem of calculus in any way? Thanks!
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  2. #2
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    Quote Originally Posted by lute View Post
    Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.
    You do know how one proves a set is a subspace?

    If \alpha is a scalar and \left\{ {f,g} \right\} \subseteq W then what can you say about \int_a^b {\alpha f + g} ?
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  3. #3
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    Quote Originally Posted by lute View Post
    Hi all,

    The question goes: Let V be the set of all continous functions on the closed interval [a,b]. Prove that the set W of continuous functions f(x) on [a,b] such that integral(f(x)dx) = 0 from a to b is a subspace of V.

    Does this involve utilizing the fundamental theorem of calculus in any way? Thanks!
    No, it involves the definition of "subspace"! If f and g are two continuous functions such that \int_a^b f(x)dx= 0 and \int_a^b g(x)dx is f+ g a continuous function such that \int_a^b(f(x)+ g(x))dx= 0? If n is a real number is nf(x) a continuouis function such that \int_a^b  nf(x)dx= 0?
    Last edited by HallsofIvy; February 7th 2010 at 01:51 AM.
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