Functional Analysis 26 | Open Mapping Theorem

[@mattetor6726]

I remember this one from analysis. Such a nice and reasonable result, but such a difficult proof!

[@亚洲亚洲]

Thank you sir.

[@mathe_ma]

Eagerly waiting for this ... Sir you makes things very easy

[@johnstroughair2816]

Are you going to make videos proving all the theorems you have introduced in the last few lectures?

[@brightsideofmaths]

Yes, of course. But not right away. There are still a lot of other theorems we should discuss first.

[@hafsasaima4163]

When r u posting the proofs

[@brightsideofmaths]

Only if some people are interested in them.

[@wesleyrm]

Well, I didn't actually catch/note all of them, but the four main ones for now are the proofs for Hahn-Banach (Geometric & Analytic), Arzelà-Ascoli, Banach-Schauder (Open mapping theorem) and Banach-Steinhaus (Uniform boundedness principle). The Baire Category Theorem, BCT (I have no idea what this is lol), can be used to prove the last two, but I read somewhere they can even be proven "more elementarily" without it... Their proofs can all be found on Wikipedia. With the exception of Hahn-Banach, all their proofs can be found on ProofWiki as well, which I prefer to Wikipedia. For Arzela-Ascoli, some sources cite different types of equicontinuity on the definition, but apparently they can be proven to be equivalent in compact metrics spaces... My course on university uses the two versions of the Hahn-Banach theorem. Also they work a lot with Convex sets, and topology. EDIT: In my course they talk about Weak topologies and all that stuff I have no idea about lol. So we see Banach-Alaoglu theorem for *-weak topology, which uses Tikhonov's theorem. Stuff like this is still way outside my reach... proofwiki.org/wiki/Projection_on_Real_Euclidean_Plane_is_Open_Mapping Even a fact on projections, which may seem simple, not even a theorem, but a counterexample, requires knowledge on Topology...

[@InfiniteStudy]

+1

[@irshadsirslectures4446]

Is this both was result OMT

[@sweety3426]

Why x----x² not open?? How you told that

[@brightsideofmaths]

Why do you think it's open?

[@SG_0312]

@sweety choose (-1, 1) then f (-1, 1) =[0, 1) which is not open.