Understanding Theorems: Open Mapping & Closed Range

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In summary, Miss. Lolitta is looking for a complete lecture on "the open mapping theorem" and "the closed range theorem" but has not found clear explanations in books. She receives a few definitions from another user, as well as links to resources for further understanding.
  • #1
Miss_lolitta
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Miss. Lolitta says:

Hello everybody here :smile:

Can someone give me a complete lecture-that has introduction & examples and explaining-for

"the open mapping theorem" and "the closed range theorem"

actually I read some books about this theorem but they weren't clear for me:bugeye:

please,help me to figure out this theorem..

Thanks in advance
 
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  • #2
Hi,

I'm not familiar with the theorems, but I can give a few definitions:

f is an open mapping if for all open sets O in the domain, f(O) is open in the range. In other words, f maps open sets to open sets.

f is a closed mapping if for all closed sets C in the domain, f(C) is closed in the range. In other words, f maps closed sets to closed sets.

I'm not sure what theorems you are talking about.
 
  • #4
thanks sooooooo much
 
  • #5

Related to Understanding Theorems: Open Mapping & Closed Range

1. What is the definition of an open mapping theorem?

The open mapping theorem is a mathematical theorem that states that a continuous linear transformation between two Banach spaces is either an open map (maps open sets to open sets) or a constant map (maps all sets to a single point).

2. What is the significance of open mapping theorem in mathematics?

The open mapping theorem has significant applications in the field of functional analysis and topology. It is used to prove other important theorems such as the inverse mapping theorem and the closed graph theorem.

3. How does the open mapping theorem relate to the concept of surjectivity?

The open mapping theorem guarantees that a continuous linear transformation between Banach spaces is surjective (onto) if and only if it is an open map. This means that every element in the target space has at least one pre-image in the domain.

4. What is the closed range theorem and how is it related to the open mapping theorem?

The closed range theorem is a mathematical theorem that states that a continuous linear transformation between two Banach spaces is onto (surjective) if and only if its range (image) is closed. This theorem is closely related to the open mapping theorem and is often used in conjunction with it.

5. Can the open mapping theorem be applied to infinite-dimensional vector spaces?

Yes, the open mapping theorem can be applied to infinite-dimensional vector spaces, as long as the spaces are complete (i.e. Banach spaces). This is one of the reasons why the open mapping theorem is so important in functional analysis, where infinite-dimensional spaces are commonly studied.

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