Lecture 4: The Open Mapping Theorem and the Closed Graph Theorem

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MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

We prove two more fundamental “theorems with names” as Casey puts it: the Open Mapping Theorem and the Closed Graph Theorem. We conclude with the notion of a Hamel basis for a vector space (finite or infinite dimensional).

License: Creative Commons BY-NC-SA

  1. MIT OpenCourseWare
  2. Closed Graph Theorem
  3. Hamel basis
  4. Open Mapping Theorem
  5. Zorn’s lemma
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Комментарии
Автор

"So let's continue with the 'big-name theorems' or the 'theorems with big names' or the 'big theorems with names'", absolutely hilarious!

travischapman
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Thanks mit for providing high quality education

coreconceptclasses
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Open mapping theorem is the most used result from functional analysis

briang.valentine
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35:37 End of proof to open mapping theorem
57:21 End of proof to closed graph theorem

oldPrince
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At 35:25, you mark the proof of the open mapping theorem as complete. I'm not sure that it is. Maybe it's my own ignorance. I've never taken any analysis or topology courses. I've only watched some lectures on YouTube, including a few from your course on real analysis.

It seems to me that you've shown that T(U) contains an open ball, namely B(b2, epsilon*delta). But I don't see how this implies that T(U) is open (I'm assuming you mean open with respect to the topology on B2 induced by the norm). This is especially confusing for me since, in Baire's theorem, it was shown that at least one of those *closed* subsets (given by the conditions in the theorem) contains an open ball. Am I missing something?

zn
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I remember when the 4 color was a lived controversy.

SSNewberry
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When would any of this be used in life?

joenissan