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

Let

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

Re: Finding subspace/continuous function

Quote:

Originally Posted by

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

Good.