# Topological vector spaces

1. Feb 12, 2009

### dori1123

Let $$C([0,1])$$ be the collection of all complex-valued continuous functions on $$[0,1]$$.
Define $$d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx$$ for all $$f,g \in C([0,1])$$
$$C([0,1])$$ is an invariant metric space.
Prove that $$C([0,1])$$ is a topological vector space

2. Feb 12, 2009

### shoehorn

No thanks. Why don't you prove it for us?

3. Feb 13, 2009

### HallsofIvy

If you have already proved that C([0, 1]) is a metrix space, and it is clearly a vector space, all you need to prove is that the functions F(f, g)= f+ g and F(a, f)= af are continuous.

4. Feb 13, 2009

### dori1123

I don't know how to show F(a,f)=af is continuous, can I get some hints please.

5. Feb 13, 2009

### Hurkyl

Staff Emeritus
Have you tried using the definitions? Or tried anything at all?

(P.S. you should always try to use the definitions when you're stuck on any problem)

6. Feb 13, 2009

### dori1123

F is continuous if for every open subset U of Y, F^{-1}(U) is open in X.
So I let U be an open subset in Y, and let (a,f) be in F^{-1}(U). Then F(a,f) is in U. then I am stuck...

7. Feb 14, 2009

### adriank

Since you've got a metric space, it might be easier to use the epsilon-delta formulation of continuity to show it.

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