Proving Banach Space Property Using Topological Isomorphism

[dirk_mec1]

Homework Statement

http://img219.imageshack.us/img219/2512/60637341vi6.png

Homework Equations

I think this is relevant:
http://img505.imageshack.us/img505/336/51636887dc4.png

The Attempt at a Solution

A topological isomorphism implies that T and T^-1 are bounded and given is that all cauchy sequences in E are convergent.

But if that's the case then by boundedness all convergent sequences are bounded in het norm of F and are thus also Cauchy seqeunces. Am I thinking in the right direction?

 

[morphism]

But if that's the case then by boundedness all convergent sequences are bounded in het norm of F and are thus also Cauchy seqeunces. Am I thinking in the right direction?

You have to prove that every Cauchy sequence in F converges. How is what you're saying doing that?

 

[dirk_mec1]

Okay so that's wrong but suppose I have a cauchy sequence in E: $$|x_n-x_m|_E < \epsilon\ , \forall n,m\geq N$$ how can I prove then that F is also Banach?

 

[morphism]

Why are you taking a cauchy sequence in E? You're supposed to prove that F is a Banach space.

 

[dirk_mec1]

Why are you taking a cauchy sequence in E? You're supposed to prove that F is a Banach space.

Because it is given that E is Banach what implies that every cauchy sequence converges.

 

[morphism]

What I meant is that it makes more sense to start with a cauchy sequence in F rather than in E. Because you want to prove that every cauchy sequence in F converges.

 

[dirk_mec1]

Can someone give me a hint? Because I've started with a cauchy sequence in F but I honestly do not see what to do next.

 

[HallsofIvy]

Let {a_n} be a sequence in F. What can you say about {T^-1 a_n}?

 

[dirk_mec1]

Let {a_n} be a sequence in F. What can you say about {T^-1 a_n}?

Let $$\{ a_n \}$$ be a sequence in F then for all $$n,m \geq N$$ we have:


$$|| T^{-1} (a_n-a_m)||_E \leq c\cdot ||a_n-a_m||_F < c \cdot \epsilon$$

So a_n is Cauchy in F.

But how do I get it to converge in F with limit a?

 

[morphism]

Look, you want to show that F is a Banach space. So you take an arbitrary cauchy sequence in F and show that it converges in F. What do we have to work with here? We know that F is isomorphic to a Banach space. Use that isomorphism.