Topological vector spaces

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

Related Differential Geometry News on Phys.org

shoehorn

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

HallsofIvy

Homework Helper
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

Staff Emeritus
Gold Member
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...

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

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving