Metric Spaces

  Рет қаралды 29,072

Dr. Will Wood

Dr. Will Wood

Күн бұрын

Пікірлер: 30
@isxp
@isxp 3 жыл бұрын
Thank you for a clear explanation and visually appealing graphics. The font, color choice, and animations are all nice to look at. You're helping an ME senior secure his Math minor with these videos
@DrWillWood
@DrWillWood 3 жыл бұрын
Thanks so much! Good luck with the numerical methods class (One of my favourite topics and very difficult but so useful!).
@蔡小宣-l8e
@蔡小宣-l8e 2 жыл бұрын
Brief and clear! Thank you. 谢谢。
@5Stars49
@5Stars49 10 ай бұрын
Lovely explanation 😮
@t.ganesh1692
@t.ganesh1692 3 жыл бұрын
Excellent video! Thank you Will!
@ibmalik84
@ibmalik84 2 жыл бұрын
your channel is really helpful!!
@mcgrewgs
@mcgrewgs 3 жыл бұрын
At around 2:10, that arrow could be bidirectional, right, since the two statements must be equivalent? I.e. d(x,y)=0 x=y If it weren't bidirectional, I could define, for example, d(x,y)=1 for all x,y and that would fulfill all the requirements.
@einstien311
@einstien311 3 жыл бұрын
Correct, the impilcation there should be bidirectional. If the distance between two elements is zero, then the two elements must be the same. Conversely, if two elements are the same element, the distance between the element and itself must be zero. If you are trying to prove some proposed function is a distance function, then you have to show forward implication and the backward implication. Interestingly, there is a distance function called the "discrete metric" (I think) that is defined as d(x,y)=0 if x=y, and d(x,y)=1 if x≠y. Essentially, the discrete metric just is a toggle that indicates whether the inputs are the same or different.
@edgarbonet1
@edgarbonet1 3 жыл бұрын
Indeed. As written, d(x, y) = 1+|x−y| would be a distance over ℝ, which is wrong.
@stipepavic843
@stipepavic843 2 жыл бұрын
subbed instantly, great videos!!!
@JordanWeitz
@JordanWeitz 3 жыл бұрын
Hrm. at 9:07 Will seems to claim a < b ==> |a| < |b|. It is true in this case, but he doesn't explain why.
@FreeAsInFreeBeer
@FreeAsInFreeBeer 3 жыл бұрын
Yes, this argument is not correct. (counter example: -3 < 2, but |-3| > |2|) An easy way to prove it is like this. These inequalities hold by definition: -|a| ≤ a ≤ |a| -|b| ≤ b ≤ |b| Add the inequalities together: -(|a| + |b|) ≤ a + b ≤ |a| + |b| We solve by cases: If a+b not negative, then: a + b = |a + b|, and so a + b = |a + b| ≤ |a| + |b|. If a+b negative, then a + b = -|a + b|, and so -(|a| + |b|) ≤ a + b = -|a + b|, multiplying with -1 gives: |a + b| ≤ |a| + |b|. So in both cases it is true that: |a + b| ≤ |a| + |b|. Now, let a = x-y, b = y-z. |x-y + y-z| ≤ |x-y| + |y-z| |x-z| ≤ |x-y| + |y-z| Done!
@zapzya
@zapzya 3 жыл бұрын
This is true if 0
@MT.632
@MT.632 2 жыл бұрын
Excellent video!!🌸
@choukriadan4706
@choukriadan4706 2 жыл бұрын
Thank you for the explanation Dr.will . But this units of so much though
@edgbaston149
@edgbaston149 2 жыл бұрын
Thank you so much You're amazing 👏💗
@Atistatic
@Atistatic Жыл бұрын
Dr. Wood i would like to ask if anyone who want understand metric spaces, knowning well multi variable calculus is a prerequisite??
@DrWillWood
@DrWillWood Жыл бұрын
I don't think so! Knowing the basics of set theory is enough to get started
@BHuman2024
@BHuman2024 3 жыл бұрын
Would you please suggest any book for learning metric space more.
@DrWillWood
@DrWillWood 3 жыл бұрын
Hi! I used Introduction to Topology by Bert Mendelson (Dover Books) for preparing this video which has a chapter on metric spaces and is a nice book in general. I've also heard positive things about Metric Spaces: iteration and application by Victor Bryant but I haven't actually looked at that book myself!
@zetacrucis681
@zetacrucis681 3 жыл бұрын
8:53 This step is invalid. You can't just take absolute values of both sides of an inequality, e.g.: -2 < 1 does not imply |-2| < |1|, i.e., 2 < 1.
@josephcampbell4877
@josephcampbell4877 2 жыл бұрын
He explains in the video that it's valid since the right hand side of the equation is non-negative
@bartomiejpotaman6973
@bartomiejpotaman6973 2 жыл бұрын
@@josephcampbell4877 the problem is that we don’t know the sign of the left side, zeta crucis is right about that one
@user-vg7zv5us5r
@user-vg7zv5us5r 2 жыл бұрын
More rigorous vectors sans direction parameter.
@alminaahmed_2552
@alminaahmed_2552 3 жыл бұрын
It's ❤
@magroubezpieczeniasp.zo.o.2137
@magroubezpieczeniasp.zo.o.2137 2 ай бұрын
d(x, y) = d(y, x) ... not in family court
@tombouie
@tombouie 3 жыл бұрын
Thks, ?????but what is the clear difference between a metric vs normed space???? en.wikipedia.org/wiki/Metric_space vs en.wikipedia.org/wiki/Normed_vector_space
@DrWillWood
@DrWillWood 3 жыл бұрын
Hi! A normed linear space (NLS) IS a metric space but the definition but made a little bit more specific to vector space. Its the property (3.) d(x,y) = d(y,x) that is changed. i.e. the distance is same in either direction. for NLS, if we reverse the direction of a vector X we multiply by -1. By property 3 of a NLS we have ||-1X|| = |-1| ||X|| = ||X|| and so its the same with NLS that direction doesn't matter. But we also want that intuitive property of vectors built in that 2 times a vector should by twice as long and so on so that's we why have ||aX|| = |a| ||X|| for constant a in general. The way I think about it is more simple: that a NLS is a metric space that happens to also be a vector space!
@tombouie
@tombouie 3 жыл бұрын
@@DrWillWood Ooooooh, thks-yous ever so much.
@Joffrerap
@Joffrerap 3 жыл бұрын
distance has 2 inputs and norm only has 1 so that's one big difference already.
Normed Linear Spaces | Introduction,  L1 and L2 Norms
13:56
Dr. Will Wood
Рет қаралды 27 М.
The applications of non-euclidean distance | Metric Spaces
18:43
Family Love #funny #sigma
00:16
CRAZY GREAPA
Рет қаралды 46 МЛН
When mom gets home, but you're in rollerblades.
00:40
Daniel LaBelle
Рет қаралды 146 МЛН
СОБАКА ВЕРНУЛА ТАБАЛАПКИ😱#shorts
00:25
INNA SERG
Рет қаралды 3,6 МЛН
The Singing Challenge #joker #Harriet Quinn
00:35
佐助与鸣人
Рет қаралды 33 МЛН
Padé Approximants
6:49
Dr. Will Wood
Рет қаралды 442 М.
What is a Topological Space?
9:41
Infinite Dimensions
Рет қаралды 51 М.
Approximating Functions in a Metric Space
7:46
Dr. Will Wood
Рет қаралды 56 М.
The Concept So Much of Modern Math is Built On | Compactness
20:47
Morphocular
Рет қаралды 437 М.
Weird notions of "distance" || Intro to Metric Spaces
12:31
Dr. Trefor Bazett
Рет қаралды 92 М.
What is Jacobian? | The right way of thinking derivatives and integrals
27:14
Can you solve this Cambridge Entrance Exam Question?
24:48
Higher Mathematics
Рет қаралды 797 М.
The deeper meaning of matrix transpose
25:41
Mathemaniac
Рет қаралды 387 М.
what is i factorial?
7:56
blackpenredpen
Рет қаралды 313 М.
Metric Spaces 1: Definition and Examples
54:46
Math at Andrews University
Рет қаралды 6 М.
Family Love #funny #sigma
00:16
CRAZY GREAPA
Рет қаралды 46 МЛН