Is U a Subspace of F([a, b]) for Real-Valued Functions with Certain Conditions?

[theRukus]

Homework Statement

Is U = {f E F([a, b]) | f(a) = f(b)} a subspace of F([a, b]), where F([a, b]) is the vector space of real-valued functions de ned on the interval [a, b]? (keep in mind that in the definition of U, the E means belonging to.. I couldn't find an epsilon character)

Homework Equations

The Attempt at a Solution

I know I have to check the following two closure axioms:

Check that C belongs to U where C = A + B and A, B belong to U

Check that C belongs to U where C = kA and A belongs to U and k belongs to real numbers.

My issue is that I just don't know how to portray an example that belongs to U. I hope I'm making some sense here.. I just need to know how A, B should look for my closure axioms.

Thanks

 

[Mark44]

Homework Statement

Is U = {f E F([a, b]) | f(a) = f(b)} a subspace of F([a, b]), where F([a, b]) is the vector space of real-valued functions de ned on the interval [a, b]? (keep in mind that in the definition of U, the E means belonging to.. I couldn't find an epsilon character)

Homework Equations

The Attempt at a Solution

I know I have to check the following two closure axioms:

Check that C belongs to U where C = A + B and A, B belong to U

Check that C belongs to U where C = kA and A belongs to U and k belongs to real numbers.

My issue is that I just don't know how to portray an example that belongs to U. I hope I'm making some sense here.. I just need to know how A, B should look for my closure axioms.

Thanks

The things in U are functions, so let's assume that f and g belong to U. What does it mean for a function to belong to this set?

 

[Dick]

Well, A(a)=A(b) and B(a)=B(b). Show C(x)=A(x)+B(x) satisfies C(a)=C(b). That's all they're asking for the first closure. That isn't so hard, is it?

 

[theRukus]

Not at all! Thanks so much