Finding subspace/continuous function

Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?

Re: Finding subspace/continuous function

Quote:

Originally Posted by

**steph3824** Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?

You have to prove the subspace axioms.

U is nonempty since $\displaystyle f\in U$

Let $\displaystyle f,g\in U$

$\displaystyle (f+g)(1/2)=f(1/2)+g(1/2)=f(1)+g(1)=(f+g)(1)$

How about the last axiom now.

Re: Finding subspace/continuous function

Would it be (af)(1/2) = a·f(1/2) = a·f(1)?

Re: Finding subspace/continuous function

Quote:

Originally Posted by

**steph3824** Prove that the set U={f ∈ C([0,1]): f(1/2)=f(1)} is a subspace of C([0,1]).

I am not sure how to do this. I know what the necessary conditions are for a subspace but I can't quite figure out how to show them given a continuous function. Help please?

I don't suppose you know the fact that if $\displaystyle T:V\to W$ is a linear equation, then $\displaystyle \ker T$ is a subspace of $\displaystyle V$. Well, I'm sure you can note that $\displaystyle C[0,1]\to\mathbb{R}:f\mapsto f(1)-f(\tfrac{1}{2})$ is a linear transformation and $\displaystyle \ker f=U$.

Re: Finding subspace/continuous function

Quote:

Originally Posted by

**steph3824** Would it be (af)(1/2) = a·f(1/2) = a·f(1)?

Good.