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Homework Help: Vector Space Problem

  1. Sep 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Do functions that vanish at the end points x=0 and x= L form a vector space? What about periodic functions obeying f(0)=f(L)?How about functions that obey f(0)=4

    2. Relevant equations
    3. The attempt at a solution

    We consider functions defined at 0<x<L.We define scalar multiplication by a simply as af(x) and addiion as pointwise addition: f(x)+g(x) at every point x.The null function is zero everywhere and the additive inverse is -f(x).

    First kind of functions satisfy closure,commutativity and associativity of addition.They are OK with scalar multiplication.They have in their set the null element: a null function which is zero everywhere.They also contain -f(x).So,they can form a vector space.

    The third kind of functions: obeying f(0)=4 exclude the existence of null function (which is zero everywhere) and the existence of -f(x)...This set is also not going to form a vector space.

    I think the periodic functions will form a vector space only if null function is considered to be a periodic function of arbitrary period.For this kind of functions,we are given, f(0)=f(L). It seems that other conditions concerning the closure are satisfied.

    Please tell me if I am missing something.
  2. jcsd
  3. Sep 13, 2008 #2


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    Science Advisor
    Homework Helper

    Seems fine. The null function (and other constant functions) ARE periodic.
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