1. ## Open map theorem

Hello!
I wanted to ask something about functional analysis but i looks that this topic doesn't exist. I hope it is ok to publish it here.
Well, I wanted to proove that:
Let X and Y be two Banach spaces and "f" a map between the, such tahta it is linear, continuous and surjerctive. Then:
If K is a compact subset of Y then there exists L which is a compact subset of X such that f(L)=K.

Well, I know I have to apply that because of the open map theorem T is an open map but I am not sure about what to do.
Any ideas???
Thank you very much!!

2. ## Re: Open map theorem

Originally Posted by Lolyta
Hello!
I wanted to ask something about functional analysis but i looks that this topic doesn't exist. I hope it is ok to publish it here.
Well, I wanted to proove that:
Let X and Y be two Banach spaces and "f" a map between the, such tahta it is linear, continuous and surjerctive. Then:
If K is a compact subset of Y then there exists L which is a compact subset of X such that f(L)=K.

Well, I know I have to apply that because of the open map theorem T is an open map but I am not sure about what to do.
Any ideas???
Thank you very much!!
perhaps consider the mapping of the complements of L and K