Topological vector spaces

  • Thread starter dori1123
  • Start date
11
0
Let [tex]C([0,1])[/tex] be the collection of all complex-valued continuous functions on [tex][0,1][/tex].
Define [tex]d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx[/tex] for all [tex]f,g \in C([0,1])[/tex]
[tex]C([0,1])[/tex] is an invariant metric space.
Prove that [tex]C([0,1])[/tex] is a topological vector space
 
422
1
No thanks. Why don't you prove it for us?
 

HallsofIvy

Science Advisor
Homework Helper
41,711
876
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.
 
11
0
I don't know how to show F(a,f)=af is continuous, can I get some hints please.
 

Hurkyl

Staff Emeritus
Science Advisor
Gold Member
14,845
17
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)
 
11
0
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...
 
534
1
Since you've got a metric space, it might be easier to use the epsilon-delta formulation of continuity to show it.
 

Related Threads for: Topological vector spaces

Replies
2
Views
2K
Replies
2
Views
2K
  • Posted
Replies
9
Views
4K
  • Posted
Replies
4
Views
3K
  • Posted
Replies
8
Views
4K
  • Posted
Replies
2
Views
6K
Replies
7
Views
948
  • Posted
2
Replies
33
Views
7K

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
Top