An open mapping is not necessarily a closed mapping in functional analysis

[adityab88]

We know that a linear operator $$T$$:X$$\rightarrow$$Y between two Banach Spaces X and Y is an open mapping if $$T$$ is surjective. Here open mapping means that $$T$$ sends open subsets of X to open subsets of Y.
Prove that if $$T$$ is an open mapping between two Banach Spaces then it is not necessarily a closed mapping, i.e. there could exist a closed subset of X that maps to a subset of Y which is not closed.

In other words, give a counter example. Being new to functional analysis, this has made me scratch my head..

 

[fourier jr]

consider $$\inline{\{(x,y) | xy=1\} \subseteq \mathbb{R}^2}$$ and the projection onto the first component $$\inline{(x,y) \mapsto x}$$, which is your open mapping

 

[Urmi Roy]

Hi,
I dug this thread up and I thought it might be useful to get some of my own problems cleared here itself,instead of starting a while new thread...

1. Firstly,I don't understand why we have to distinguish open and closed intervals ...an interval is said to be open when the boundary points are not considered...but every region has to have a boundary! For example a circle has a boundary...even if we don't consider the points on the boundary,there is another boundary just within the original one!

2.Why is the neighbourhood of a point always an open interval?

3.In complex analysis,why is a domain defined exclusively as an open interval...whereas a region is a domain with the boundary...is there any reason why we work with only open intervals in complex analysis?

(An additional criterion for domain is that it has to be open and connected)...is this connectedness necessary simply to ensure that the function is determined everywhere inside it?)

4. Where do we use the concept of half planes?They are defined in Kreysig,but we don't seem to use them very often.

Thanks in advance.

 

[adityab88]

Hi,
I dug this thread up and I thought it might be useful to get some of my own problems cleared here itself,instead of starting a while new thread...

1. Firstly,I don't understand why we have to distinguish open and closed intervals ...an interval is said to be open when the boundary points are not considered...but every region has to have a boundary! For example a circle has a boundary...even if we don't consider the points on the boundary,there is another boundary just within the original one!

2.Why is the neighbourhood of a point always an open interval?

3.In complex analysis,why is a domain defined exclusively as an open interval...whereas a region is a domain with the boundary...is there any reason why we work with only open intervals in complex analysis?

(An additional criterion for domain is that it has to be open and connected)...is this connectedness necessary simply to ensure that the function is determined everywhere inside it?)

4. Where do we use the concept of half planes?They are defined in Kreysig,but we don't seem to use them very often.

Thanks in advance.

Firstly, I strongly suggest you study topology. You won't be able to understand such concepts without knowledge of that course, and it is a crucial course in your study of mathematics. Just read any introductory book. Your question is very basic (I think it will also be answered in a standard rigorous analysis class/book).

1. An open set, in your setting, does not have a "boundary" by your sense of the word. Take the open interval (-1,1) in R, the real numbers. Now I ask you to give me a "boundary". Say you give me the set {-1, 1} as the only boundary points. However, these points are not in the set, so don't qualify. If you choose {-1+e, 1-f} as the boundary for any e,f>0, we can always challenge that claim by putting {-1+e/2, 1-f/2} as the boundary, which is closer to the edges of the open interval. So there is no boundary. If you find me any boundary, I can prove it to you that it is not a boundary. Not the case with closed interval say [-1, 1]. Boundary is clear to be {-1,1}. Also it is difficult for me to explain this to you beucase you are using some layman terms and are not used to standard mathematical jargon (i think so). In that case, I again refer you to topology book. Get your definitions right of terms. What is a open set really? What is a closed set? What is a boundary? Did you know, that there are topological spaces where your current intuition of open and closed sets is completely different? Indeed, there are spaces where every single set is both open and closed.

2. Thats the definition of a neighborhood. There's no proof of this. You can define a neighborhood in any way you like, this is just the chosen definition.

3. If you study some advanced complex analysis you will notice that the most interesting properties will be true for a function f if the domain of definition is an open set. Being connected also tends to help. That is why we use open (and connected) sets. Again, matter of definition.

4. Weird question this. I think its just a useful definition of something in mathematics. It can crop up in all sorts of places. You should not be worried that you haven't seen it yet or anything. I suggest you continue to read Kreysig's book(?) to find some places where the term is used. Again, in mathematics, you can define anything. In short, many of your questions are to do with definitions of terms. You will understand as you study more mathematics the difference between a definition and a theorem. It sounds simple, but you will only gain a good grasp of this once you get the so-called "mathematical maturity".

FYI, I have seen the use of half-planes extensively in Computational Geometry.

 

[Urmi Roy]

Thanks adityab88,I'm sure that all my confusions are due to my ignorance of these higher concepts!
I'll take your suggestion and read further.

Thanks for the clarifications anyway!