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Functions as a vector space

  1. Sep 12, 2008 #1
    1. The problem statement, all variables and given/known data
    "Consider all functions f(x) defined in an interval 0[tex]\leq[/tex]x[tex]\leq[/tex]L. We define scalar multiplication by a simply as af(x) and addition as pointwise addition: the sum of two functions f and g has the value f(x)+g(x) at the point x. The null function is simply zero everywhere and the additive inverse of f is -f.
    Do functions that vanish at the end points x=0, x=L form a vector space? How about periodic functions obeying f(0)=f(L)? How about functions that obey f(0)=4? If the functions do not qualify, list the things that go wrong"

    2. Relevant equations
    Definition of vector space, features of vector sum and scalar multiplication and some axioms.

    3. The attempt at a solution
    This is a bit confusing to me so I'll be glad if some can clarify this for me.
    Some of the properties seem to agree with the definition of vector space (scalar multiplication, null function, the inverse of f) . What I find confusing is the argument x. After the sum of two functions for example I no longer get a new one just because the sum is evaluated at the point x, thus I no longer get an abstract object like a matrix or some sort of vector but I get a number. If on the other hand the sum would be defined through all x that would seem to make more sense as we would get another function.
    I don't see a problem with functions vanishing at the ends of the interval, they still might represent a vector space as long as the definition of the sum is changed ( as explained earlier). However, I do think there is a problem with f(0)=4 since we do get a number.

  2. jcsd
  3. Sep 12, 2008 #2


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    It's just a matter of checking the properties of a vector space. For example, in the first one:
    1) Does the null function vanish at the end points?
    2) If f and g vanish at the end points, does f + g?
    3) If f vanishes at the end points, does af (for a some number)?
  4. Sep 13, 2008 #3
    thanks. Yes, this far I understand. But what about my arguments?
  5. Sep 13, 2008 #4


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    The sum of two functions is again a function.
    Suppose you have a vector space V of functions on some set D. That is, if f and g are elements of V, then they are both functions on D.
    Now we want to "add" these functions and call the result "f + g" (we might as well have written it "s"). Since "f + g" is a function, we need to specify its values, that is: give a prescription. Now we say that the "f + g" is the function, which takes the value f(x) + g(x) in every point x of D.

    For example, if [tex]f = (x \mapsto x^2), g = (x \mapsto \sqrt{x})[/tex] then f + g is the function [tex]x \mapsto x^2 + \sqrt{x}[/tex]. If [tex]h = (x \mapsto -x^2)[/tex], then f + h is the null function [tex]0 = (x \mapsto 0)[/tex] (note: the "0" denoting the null function is a function, it assigns the number 0 from the codomain to every x in D).
  6. Sep 13, 2008 #5
    Oh, this becomes clearer or at least I hope so. But I am still confused about the part where I have to specify the values of the function. I realise that otherwise I won't be able to get the null function but other than that I don't see a problem
  7. Sep 13, 2008 #6
    I do not know how this fellow also got stuck with the same problem...however,if people are interested,my post may also be referred to...
  8. Sep 13, 2008 #7
    :) Thanks neelakash. I'll look into it
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