Vector Space Problem: Does it Form a Vector Space?

[neelakash]

Homework Statement

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

Homework Equations

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.

 

[Dick]

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