Lecture 1: Motivation, Intuition, and Examples
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- Опубликовано: 9 фев 2025
- MIT 18.S190 Introduction To Metric Spaces, IAP 2023
Instructor: Paige Bright
View the complete course: ocw.mit.edu/co...
RUclips Playlist: • MIT 18.S190 Introducti...
Introduction to the theory of metric spaces-the tool used to generalize the theory of real analysis from the real numbers to more abstract settings. Includes many classic examples of metric spaces and key terminology.
This video has been dubbed using an artificial voice via aloud.area120.... to increase accessibility. You can change the audio track language in the Settings menu.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
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We encourage constructive comments and discussion on OCW’s RUclips and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at ocw.mit.edu/co....
MIT 18.S190 Introduction To Metric Spaces, IAP 2023
Instructor: Paige Bright
View the complete course: ocw.mit.edu/courses/18-s190-introduction-to-metric-spaces-january-iap-2023/
RUclips Playlist: ruclips.net/p/PLUl4u3cNGP613ULTyHAqz04niYf722x7S
Introduction to the theory of metric spaces-the tool used to generalize the theory of real analysis from the real numbers to more abstract settings. Includes many classic examples of metric spaces and key terminology.
This video has been dubbed using an artificial voice via aloud.area120.google.com to increase accessibility. You can change the audio track language in the Settings menu.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu
Support OCW at ow.ly/a1If50zVRlQ
We encourage constructive comments and discussion on OCW’s RUclips and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at ocw.mit.edu/comments.
Hi and thank you for this course
Thank you very much for this course :-)
La verdad felicitarlos por el increíble trabajo de hacer más accesible este conocimiento para los estudiantes alrededor del mundo que hablan español.
Las definiciones que se dan de sucesión y de continuidad son incorrectas. Lo digo no para criticar a quien está exponiendo sino para prevenirlos a ustedes. Para aprender no hay como los libros.
Thank you MIT! After I finished 18.100A, I am waiting for a course that can introduce the metric space. Now I can watch this and 18.102!
Whooo... Man, what are you going to do when you have completed this lectures sir? Actually I'm quite curious!
@@tejaswithme3713 same?
Just started but i can already tell im gonna enjoy this. Love your explanation of the def of a metric space.
a great lecture by Paige! hope you go onto accomplish great things in life !
Crystal clear.
I'm watching this with some background in topology (via Stone spaces and logic, like Steven Vickers' Topology via Logic) but not having taken real analysis. It is interesting to see the generalization from calculus definitions on R^n to a more general setting but still not quite as general as topological spaces. It provides me a bit of the bigger picture, for when I finally dive into real analysis I will know where things are going, and how I will eventually get to where I want to go with topological spaces, locales and toposes.
Les deseo lo mejor a las personas que estan viendo este curso!
Very nice explanations, very nice examples. Great work, thank you, Paige 👏👏
52:00 if the proof is made by contradiction, the hypothesis is I_1(f,g)=0 on [0,1] and one, by negating the thesis, supposes g different from f on [0, 1]. Since f, g are both continuous there exists a ball on which their difference cannot be zero, hence the integral on that ball contained in [0,1] is not zero and this leads to a contradiction.
omg yes. so glad this came out
This kid got some serious style!
31:05, since the functions are continuous and the set is compact, the supremum is a maximum because of Weierstrass theorem.
Thank you MIT ❤
Muito obrigado, MIT. Tenho aprendido bastante com o vosso canal.
tomei um susto com a dublagem do Google Translate hahaha mas ficou bom mesmo.
Ustedes tienen cursos de más alto nivel en el impa de Brasil.
38:00 in case the first part of definition was taken away, then we would no more have a metric, because definitness wouldn't hold anymore. We would have a semi-metric, or to be more accurate a semi-norm, since also homegeneity hods.
Wow genial, felicitaciones.
A sequence is not a "Bijection" between natural numbers and real numbers
Yes. I bijection can't even exist if you ask Cantor.
I like the word “y‘all“ 😂
Me too, it's a completely valid contraction. The only reason it's frowned on is coastal elitism.
@@chrstfer2452what
Publish a course in Continuum Mechanics and differential geometry
hi!
plz add a courses on 18.03 and number theory
FANTASTIC Penmanship.
I downloaded this video. Unfortunately the video downloaded with an artificial voice in Spanish, even though I have set the audio track language to English...
If there is a way to download the video with the original English voice, then please let me know.
You can download videos from the course on MIT OpenCourseWare at: ocw.mit.edu/courses/18-s190-introduction-to-metric-spaces-january-iap-2023/ (there's a link under each video) and on the Internet Archive at: archive.org/details/MIT18.S190IAP23. Best wishes on your studies!
@@mitocw Many thanks, for your response and proposal. I will try it.
Deberían también subir la versión original :(
Can I expect more? 😍
This teacher is great but unfortunately the camera does not show the black board correctly.because looking at the teacher instead of looking at the board1
These videos are filmed by cameras with automatic motion tracking
@@nagarajuchukkala9538 whatever the case, his point stands
Why is the condition that d(x,y)=0 if and only if x=y called positive definite? Is it related to any other term?
I had a small conversation with ChatGPT about what the term "positive definite" means in this context, and more generally lol. Dunno if I would trust the responses ofc, and it sorta waffled back and forth on me a bit, but here are some interesting parts. It further broke up the 2nd point in the definition of a metric (positive definite) into both : " 1) Non-negativity: For any two points x and y in the space, the distance function should be non-negative, meaning d(x, y) ≥ 0." and "2) Identity of indiscernibles: The distance between any two distinct points should be zero if and only if the points are the same. In other words, if d(x, y) = 0, then x = y". The non-negative part is fairly obvious (positive), but I asked if the "definite" part was a reference to this identity of indiscernibles requirement, and it said "In the standard definition of a metric, the properties I mentioned earlier-non-negativity, identity of indiscernibles, symmetry, and the triangle inequality-are all part of the complete definition. When the term "positive definite" is used, it is often meant to encompass these properties collectively. If the lecture you are watching focuses specifically on the non-negativity aspect of the metric, it is possible that the phrase "positive definite" is being used in a more limited sense to emphasize only that property. However, in a broader mathematical context, "positive definite" typically refers to the entire definition of a metric, including all the necessary properties." So idk.. to me, definiteness seems to be the portion about d(x,y) = 0 iff x=y, because you want it to be definite that two things having the same quality (that of being 0 "distance" apart) is only possible if they are actually the same thing, and that if two things are the same thing they also share that quality. In other words, we want to exclude the things that defy this property and are in the same spot but different things. idk if this helps, gl!
@@tommyhopkins6431 I like the name identity of indiscernibles. When I mentioned other terms, I was thinking if positive definiteness as you might find on Wikipedia's listing of those terms: en.wikipedia.org/wiki/Positive_definiteness is related to this property at all.
I was also asking about it because I wasn't sure about why we wanted to include the nonnegativity of d(x,y) explicitly, since it's redundant in the sense that we are defining a metric as a function from X times X to the nonnegative reals anyways.
It seems like ChatGPT can't decide between whether positive definite is supposed to encompass something about multiple properties or one property of the metric. For now I will just take it to mean the condition as stated: that the metric is nonnegative and that the distance between two points in the set is 0 if and only if the two points are actually the same point.
d(x,y) = 0 iff x=y (reflexivity) together with the condition that d(x,y) > 0 for x =/= y is what's called positive definiteness. It just means "definitely positive".
20:00 f is continuous AT POINT x if...
The definition of continuous function appears to me to be a uniformly one.
Que bien ... !
a distance between two points is the shortest i.e. min, why d(x,z) = max|x-z|? And another. The distance from the unnamed vector to the horizontal plane (the dashed line) is the shortest, i.e. min, I mean any other path is larger than the dashed line path. Where am I losing the idea that the lecturer is trying to convey? Sorry
d is any arbitrary function that satisfies the given conditions. Some examples, I think Paige uses all of these - 1) Euclidean distance, the one you know from basic geometry 2) Manhattan distance, the componentwise sum of the absolute differences, so-called because it would be the distance in blocks you would travel in a city laid out in a grid like Manhattan in NY City, to get from one intersection to another. This generalizes to n-dimensions 3) the max of any of the component distances of the Manhattan distance. See Chebyshev distance on wikipedia.
Note that in the definition of d there is no restriction that it be a minimum.
I can't not go any competitive coding and any type contest
As far as I know, most freshman courses are taught by students. The faculty is too busy doing research, writing articles and doing grant work to give lectures. I wouldn't know bc I went to community College and all my lectures were given by ppl with Master's degrees.
1:00
🙏💛
Hi, esto en q se aplica explicar x fa
saludos desde colombia
Why do we definine metrix from XxX -> R?
Why R?
Why not a totally ordered Ableian group, or a field?
How can I apply undergraduate MIT class 11 student love from Bangladesh I am a autistic boy so I what do wat to do
The definition of sequence is wrong! Happens around 16:33. It doesn’t have to be a bijection! What about constant sequences?! Wow. They teach the wrong stuff at MIT.
I was expecting this course by one of mit faculty, not student
I can't understand your comment
@@sebon11 Think about it, it will come to you.
If MIT posted the lesson on their channel the student has their endorsement
Do you feel a 3rd year MIT student whom has taught the class for the last 2 years lacking sufficient knowledge?
It says that in the title...
More chalk on the blackboard in the computer age. My Walmart App is far more sophisticated and they deliver to my front door.
Mathematicians have always and will continue to use chalk on the blackboard. It is effective and has always been used. Get used to it.
And while we're talking about sophistication, like you could understand a single thing from this lecture, ha!
@@x0cx102 Eleven years on RUclips and only 10 subscribers -- that tells me how much your opinion is worth.
And what is wrong with chalk?
@@x0cx102 chalk irritates my eczema, I'd much rather they just use whiteboards or smartboards
Can't u do it on a tab or sth. The work is so scattered. Ur tutoring people online not in classroom