(adsbygoogle = window.adsbygoogle || []).push({}); 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.

thanks

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# Homework Help: Functions as a vector space

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