Proving Closure of X/Y in Normed Spaces

[wii]

Hi there,

Could you please help me in how to prove the following :

If Y is a closed linear subspace of a normed space X, then

if X is complete ==> X/Y is complete.

Cheers,
W.

 

[Landau]

Yes I can. But I won't unless you put some work into it.

 

[wii]

I have an idea but I'm not sure if it is right or wrong :

Let $$[x_n]$$ be a Cauchy sequence in $$X/Y$$ where $$[x_n]=\{x_n + y | y \in Y\}$$ that means the norm of $$[x_n] , [x_m]$$ in this sequence is less than $$\epsilon$$ for each $$m, n > N$$

so we want to use completeness of $$X$$, but how?

If $$[x_n]$$ is a Cauchy sequence in $$X/Y$$ ,then does that mean $$x_n$$ is a Cauchy sequence in $$X$$ ?

If so, then suppose $$x_n$$ is a Cauchy sequence in $$X$$ converges to $$x \in X$$, then $$[x_n]$$converges to $$[x] \in X\Y$$?

I am confused :s I think there is some thing missing :\

 

[micromass]

Could you also state which norm you put on X/Y?

If $$[x_n]$$ is a Cauchy sequence in $$X/Y$$ ,then does that mean $$x_n$$ is a Cauchy sequence in $$X$$ ?

This is not true. However, you may find representatives of $$[x_n]$$ that do form a Cauchy sequence. I mean: it is possible to find $$y_n\in [x_n]$$, such that $$(y_n)_n$$ does form a Cauchy sequence.
But it is in general not true that $$(x_n)_n$$ is a Cauchy sequence...

 

[wii]

|| x+Y|| = \inf _{y\in Y} ||x+y||

where ||x+y|| is the norm that defines on $$X$$.

I've already proved that ||x+Y|| defines a norm on X/Y.

Thanx in advance.

 

[Landau]

Given a Cauchy sequence ([x_n])_n in the quotient X/Y, find a subsequence ([x']_n)_n such that ||[x']_n-[x']_m||<2^{-n} for all n. From this, construct a Cauchy sequence in X.