Thread:

1. 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?

2. Re: Finding subspace/continuous function

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 $f\in U$

Let $f,g\in U$

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

How about the last axiom now.

3. Re: Finding subspace/continuous function

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

4. Re: Finding subspace/continuous function

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

5. Re: Finding subspace/continuous function

Originally Posted by steph3824
Would it be (af)(1/2) = a·f(1/2) = a·f(1)?
Good.
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