Vector Space Requirements for Functions that Vanish at x=0 and x=L

[ercagpince]

Homework Statement

Do functions that vanish at the end points x=0 and x=L form a vector space ? How about periodic functions obeying f(L)=f(0)? How about functions that obey f(0)=4 ?
If the functions do not qualify , list the things that go wrong .

Homework Equations

All axioms defining the vector space .

The Attempt at a Solution

I tried to sketch some functions collapsed on point x=L and x=0 and superposed them . But , I don't know how to make a pointwise addition of 2 functions which I don't know the explicit forms .
I guess functions mentioned on the problem are violating the closure feature of a vector space.

I got really confused

 

[anathema]

Hello,

Thank you for your question. The functions that vanish at the end points x=0 and x=L do not form a vector space. This is because they do not satisfy the closure property, which states that the sum of any two elements in the vector space must also be in the vector space. In this case, the sum of two functions that vanish at the end points may not necessarily vanish at the end points, thus violating the closure property.

Similarly, periodic functions obeying f(L)=f(0) also do not form a vector space. This is because they do not satisfy the closure property as well. The sum of two periodic functions may not necessarily be periodic, thus violating the closure property.

Functions that obey f(0)=4 also do not form a vector space. This is because they do not satisfy the zero vector property, which states that there must be an element in the vector space that behaves like the additive identity (i.e. adding it to any element in the vector space does not change the element). In this case, the function f(x)=4 does not have this property, as adding it to any other function will change the value of the function at x=0.

Overall, these functions do not form vector spaces because they do not satisfy the axioms defining a vector space, such as the closure property, existence of a zero vector, existence of additive inverses, and distributivity properties. I hope this helps clarify your confusion. Please let me know if you have any further questions.

 

[ogb p]

I would first clarify the definition of a vector space. A vector space is a mathematical structure that satisfies certain axioms, including closure under addition and scalar multiplication. In the context of functions, this means that the sum of two functions and the scalar multiple of a function should also be a function within the same vector space.

In the case of functions that vanish at the end points x=0 and x=L, these functions do not form a vector space. This is because the sum of two functions that vanish at the end points may not necessarily vanish at the end points. For example, the sum of two functions f(x)=x and g(x)=1-x would not vanish at x=0 and x=L, violating the closure property.

Similarly, for periodic functions obeying f(L)=f(0), these functions also do not form a vector space. This is because the sum of two periodic functions may not necessarily be a periodic function. For example, the sum of two periodic functions with different periods would not be a periodic function.

For functions that obey f(0)=4, these functions also do not form a vector space. This is because the scalar multiple of a function with this property would also not necessarily obey f(0)=4. For example, multiplying a function f(x)=x by a scalar 2 would result in a function that does not satisfy f(0)=4.

In summary, the functions mentioned in the problem do not form vector spaces because they do not satisfy the closure property, which is one of the defining axioms of a vector space.