Functional Analysis: Weak convergence lecture 1 - Oxford Mathematics 3rd Year Student Lecture

This is the first of three lectures on the topic of weak convergence we are showing from our 'Functional Analysis' 3rd year course. The lectures build upon results on Banach spaces and their duals that are covered earlier in the course.

In this lecture Melanie explains why familiar arguments from finite dimensional analysis break down in infinite dimensional spaces since bounded sequences cannot be expected to have convergent subsequences, so she introduces an alternative weaker notion of convergence.

All first and second year lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.

[tejasnatu]

Fantastic lecture Prof Melanie. I was struggling a bit in understanding weak convergence and related topics. However you have explained these things extremely well. Look forward to the next two lectures. Thanks so much.

[scottychen]

She bring ups Heine - Borel: that’s a comment on the topological nature of of these TVS’s…


Thats very intriguing…


I’d say the most identifying feature between infinite ‘dimensional’ and finite ‘dimensional’

Is the result of diagonalization:

In finite dimensional linear algebra, the most attractive property of a matrix is that it be (Graham-Schmidt) diagonalizable, using the familiar construction (Homotopy type theorists close your eyes) as proof.

The second order ‘perturbation’ of desirability is that it be ‘Jordan-diagonalizable’ into jordan blocks which invokes rather involved results on nilpotent matricies…


The Hilbert-Schmidt result is of a different nature. And has the peculiar property of a limit of specifically designated eigenvalues tending to 0.

This is the main feature of distinction between finite and infinite to me. Regarding the ‘dimension’ of a TVS

She’s more of a topological thinker

[scottychen]

(Comment)

Your emotionality is greatly centered upon the Hanh Banach theorem: a theorem about extending a dual operator’s domain without changing the operator’s norm.

This particular corollary (and officially there’s many different results that go under the name ‘Hanh Banach Theorem’ e.g. in Kolmogorov there’s 2 results he calls Hanh Banach) could be more succinctly put, as truly, a result concerning the dual operator as opposed to a fancy propositional event, that the operator, f, is an injection.


Since dual operators are characteristically thought of as surjective,

One has the concepts of ‘bijection’ and because these spaces have an algebraic structure, namely one of a vector space: TVS,

‘Isomorphism’

Associated to the corollary.

This should reflect in the lectures following

@17:22


If not, the concepts of weak topologie and strong topologies should be invoked to distinctly isolate any topologies being done with this fine algebraic perspective.


The concept of weak convergence is canonically tied to the concept of weak topologie explicitly:


Kolmogorov 20.2


My criticism of your attitude is that the concepts of strong and weak here make any sense whatsoever as a function of ones desire here….

That should be made explicitly clear, in the course, if this attitude makes sense.

This naming scheme isn’t creative: that was an example of sarcasm.

Of an irritatingly counter-productive nature

This is an excellent lecture however 🌞

[saaqibmahmood]

I am a visually impaired student. I am able to read text well enough only if the writing is in white and the background is dark, just as it is the case with the maths videos available om the MIT OpenCourseware. This Oxford mathematics lecture course would have been so useful to me had the instructor been writing on a blackboard using white chalk or marker. My disability prevents me from benefitting adequately enough from this such a wonderful effort!

[Im_MarkS]

Philosophy explained in maths? Do i understand it ?
Im just a plaster worker from the Netherlands trying to understand

[akumar]

Stumbled onto this page, it's like listening to a foreign language, what the heck is being said 😮

[RaviShah-tsjj]

Hello team members, weight define a things weight it day or night which show-+, or inculcated shadow weight -+ define in metrology, ,

[edernollivier]

I am interested in the weak convergence but it seems not clear, with undefined suites. Only with function it looks like easy.

[KushLemon]

What a joke of a "blackboard".

[vegetossgss]

excellent but french math class prépa are better than every school in UK or the US in pure math

[Studentcrazy]

Every thing is proof based maths in under grad