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

In summary, C([0,1]) is a collection of complex-valued continuous functions on [0,1], and is defined as an invariant metric space with the metric d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx. It is also a topological vector space, and to prove this, it suffices to show that the functions F(f,g)= f+ g and F(a,f)= af are continuous. This can be shown using the definitions of continuity or the epsilon-delta formulation.
  • #1
dori1123
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
 
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  • #2
No thanks. Why don't you prove it for us?
 
  • #3
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
I don't know how to show F(a,f)=af is continuous, can I get some hints please.
 
  • #5
dori1123 said:
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)
 
  • #6
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
Since you've got a metric space, it might be easier to use the epsilon-delta formulation of continuity to show it.
 

1. What is a topological vector space?

A topological vector space is a vector space that is equipped with a topology, which is a mathematical structure that defines the concept of continuity and convergence. This means that the space not only has the properties of a vector space, such as addition and scalar multiplication, but also has a notion of proximity and distance between points.

2. What are some examples of topological vector spaces?

Some examples of topological vector spaces include Euclidean spaces, function spaces, and spaces of continuous functions. In general, any vector space that has a topology defined on it can be considered a topological vector space.

3. What is the importance of topological vector spaces in mathematics?

Topological vector spaces play a crucial role in many areas of mathematics, including functional analysis, differential equations, and topology. They provide a framework for studying continuous transformations and have applications in areas such as physics, economics, and engineering.

4. What are the basic properties of topological vector spaces?

Some of the basic properties of topological vector spaces include the fact that addition and scalar multiplication are continuous operations, and that every topological vector space is also a topological group. In addition, the topology on a topological vector space is typically induced by a family of seminorms.

5. Are all topological vector spaces the same?

No, not all topological vector spaces are the same. The topology on a topological vector space can vary depending on the underlying vector space and the family of seminorms used to define the topology. Additionally, different topological vector spaces may have different properties and applications in mathematics and other fields.

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