Lecture 14: Basic Hilbert Space Theory

  • Опубликовано: 2 месяца назад

    MIT OpenCourseWareMIT OpenCourseWare
    подписчиков: 4,4 млн

    MIT 18.102 Introduction to Functional Analysis, Spring 2021
    Instructor: Dr. Casey Rodriguez
    View the complete course: ocw.mit.edu/courses/18-102-in...
    RUplayers Playlist: www.youtube.com/watch?v=EBdgF...
    Last time we discussed pre-Hilbert spaces (Hilbert spaces’ little brothers), and now we define Hilbert spaces (what pre-Hilbert spaces really want to be). With the notion of an inner product, we define analogous linear algebra tools for functional analysis.
    License: Creative Commons BY-NC-SA
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hitshot +4
hitshot

Why are there no lecture videos available for many courses? (e.g. Algebraic Geometry).

Месяц назад
RP 19 +3
RP 19

@MIT OpenCourseWare Fair enough, MIT. I stand corrected!! But, that was my point!! ChatGPT sounds so natural, it's hard to distinguish between an actual person and an AI. This reminds me of a scene from Will Smith's movie I, Robot where he asks the robot, "Can you write a symphony, can you turn a canvas into a masterpiece?". With technologies like DALL E and Chat GPT, we're almost there.

Месяц назад
MIT OpenCourseWare +21
MIT OpenCourseWare

Here's the chatGPT response (full disclosure—we added 'MIT' to your question): There are several reasons why there may not be lecture videos available for a particular MIT course. Some possible explanations include: The course may not have been recorded: Many MIT courses are not recorded, either because the professors do not have the resources or because they prefer to focus on in-person instruction. The course may be recorded, but the recordings are not publicly available: Some MIT courses are recorded for the benefit of students who are unable to attend in person, but the recordings are not made available to the general public. The course may be available on MIT's OpenCourseWare (OCW) platform: Some MIT courses are available on the OCW platform, which provides free access to course materials such as lecture notes, exams, and assignments. However, OCW does not typically include lecture videos. Overall, the availability of lecture videos for MIT courses can vary widely depending on the specific course and the preferences of the professors teaching it.

Месяц назад
RP 19 +4
RP 19

It sounds like a chatGPT response !! I'm not saying that it is 100%, but it sounds like that

Месяц назад
MIT OpenCourseWare +22
MIT OpenCourseWare

There are many reasons why there are no video lectures for many courses. Here are a few reasons: Reason #1: Money. It costs a lot of money to record (store, review, edit, caption, archive) lectures. We only select a limited number of courses to be recorded. We can do easily five non-recorded courses to one video recorded course. Reason #2: Time. It takes a lot of time to store, review, edit, caption, archive a recorded course. This course has minimum of 32 hours of footage. Not counting time to note IP issues, technical issues, glitches, etc. Reason #3: Participation. All courses published on MIT OpenCourseWare are voluntary. Some instructors do not want to be recorded.

Месяц назад
Aaron Robert Cattell +3
Aaron Robert Cattell

In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations.

2 месяца назад
motivao +1
motivao

Dr. Rodriguez, in minute 53:58 you mentioned that the limit as n approaches infinity, how do we know that the series converges? Thank you Professor, this has been the best Analysis class I've ever taken, I finally understand this.

Месяц назад
SHMaths
SHMaths

@David Lloyd-Jones I understood it and know it and thats a conjugate irrational thought is given with it which making it rational,skip (-1)^n,take later ones. Later ones are added to make it perfect.

Месяц назад
David Lloyd-Jones +2
David Lloyd-Jones

@SHMaths "And in finite case sum is either 0 or 1,and infinite case it's also 0 or 1." This is not sound. Infinity is not odd or even. To determine whether a number is odd or even we divide by two and look at the remainder. Infinity divided by two is infinity and doesn't have a remainder, perhaps because infinity is not a number, aka NaN. QED. Infinity? "It neither exists in nature nor provides a legitimate basis for rational thought …” David Hilbert.

Месяц назад
SHMaths +2
SHMaths

Suppose we take sum of (-1)^n,n= from 1 to 10what you get? You get 0 when n is even and 1 when n is odd. So if you add (-1)^n,if n from 1 to infinity , so you can get either 0 or 1. And in finite case sum is either 0 or 1,and infinite case it's also 0 or 1. Orthonormality is holding for infinite set. In my example you can see the logic of even odd ness you can get 0 or 1 is holding for infinite case also. (Not take it as convergent or divergent idea,take it as a sum with the property of summation holding in finite case and infinite case also,I,e giving value 0 or 1). Or you can see that R.H.S is independent of N,it will happen for any case. Its best:suppose 1/2^n be a series,then make n finite,it's convergent and less than 2,make it infinite it's also converegent and less than 2 also. That means we are able to say it because common ratio is less than 1 in finite case and infinite case also.

Месяц назад
Nick Goodwin +1
Nick Goodwin

Damn, I misread this. I thought this was the lecture on Dilbert space!

Месяц назад
Aaron Robert Cattell
Aaron Robert Cattell

very interesting stuff

2 месяца назад
Cutey +2
Cutey

Anthony Fantano covering math, I'm so happy.

Месяц назад
MD Explains +26
MD Explains

I live in Hilbert Space.

2 месяца назад
Ryan O'Rourke
Ryan O'Rourke

Are you neighbors with Dilbert?

Месяц назад
Ashish Singh +1
Ashish Singh

Well meet me on Banach spaces

2 месяца назад
Michael N/A +1
Michael N/A

More like dilbert space! Am I right?!

Месяц назад
PAUL WESLEY PREMKUMAR +1
PAUL WESLEY PREMKUMAR

Maths is strong. Problem with you is you follow some proven previous laws. Unless you break open those and see through...much gold is beyond. Live close to nature. It will teach you new maths, prodect and reality. Mit should travel with naturure for new definitions. I like it but.

Месяц назад
PLAYBOOK
PLAYBOOK

🔝🔝

2 месяца назад
Amangeldy Suyendykov
Amangeldy Suyendykov

Size of text on the blackboard is small, so no good visible.

2 дня назад
Namn Nguyễn Phương
Namn Nguyễn Phương

Me too…it in our DNATGC

2 месяца назад
Michael Fulton
Michael Fulton

Wow

Месяц назад
WastingMyLife
WastingMyLife

A better zoom on the board please

20 дней назад
MrByte editz +2
MrByte editz

Love from India

Месяц назад

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