Is C([0,1]) a Topological Vector Space?

[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

 

[shoehorn]

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

 

[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.

 

[dori1123]

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

 

[Hurkyl]

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

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)

 

[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...

 

[adriank]

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