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.

 

[Architeuthis Dux]

As a scientist, it is important to be precise and thorough in our definitions and reasoning. In this case, we need to define what we mean by "functions" and what operations are allowed. In general, a vector space is a set of objects (vectors) that can be added and multiplied by scalars, and these operations satisfy certain properties such as closure, commutativity, and associativity.

In the first case, where the functions vanish at the end points x=0 and x=L, it is not clear what kind of functions we are considering. Are they continuous? Differentiable? Without specifying the type of functions, it is difficult to determine if they satisfy the necessary properties for a vector space.

For the second case, where the functions are periodic with f(0)=f(L), we can consider the set of all continuous periodic functions on the interval [0,L]. In this case, the set would form a vector space, as it satisfies all the necessary properties. The null function would be the constant function f(x)=0, and the additive inverse would be -f(x).

For the third case, where the functions obey f(0)=4, it is not clear what operations are allowed. If we consider the set of all continuous functions on the interval [0,L], then this set would not form a vector space as it does not contain a null function (a function that is zero everywhere) and an additive inverse for each element.

In conclusion, the answer to whether these sets of functions form a vector space depends on the specific definitions and operations allowed. It is important to be clear and precise in our definitions to determine if a set satisfies the necessary properties for a vector space.