What is a metric space ?

[@drpeyam]

If you're interested in learning more about metric spaces and topology, check out my playlist: kzbin.info/aero/PLJb1qAQIrmmA13vj9xkHGGBXRMV32EKVI

[@User-gt1lu]

Could you show wich book you use?

[@drpeyam]

It’s based on Ross Real Analysis and Pugh Real Analysis, but munkres is also good

[@mathwizards6785]

The best mathematician I have ever seen..... Sir where r u from?

[@isobar5857]

Whenever I delude myself into believing I am clever I come here for some humility, and reach the inevitable conclusion that I am stupid.

[@drpeyam]

Note: In Example 5 (sequences), the max should be a sup And Example 9 is a pseudometric, so be two different sets could have distance 0

[@mohammedsalouani7672]

Hello Dr Peyam, I think also for example 6 max should be sup . Otherwise , great video !👍

[@DutchMathematician]

​@@mohammedsalouani7672 It is not wrong to replace 'max' by 'sup' in example 6: actually, in most cases it is even better, since 'max' might not be attained (although sup may exist and be finite). However, in this example, the use of 'max' in the definition of the metric is actually correct. The reason for this is that we are considering continuous functions on a closed, bounded interval. The closed and bounded interval is I = [a, b]. Let f and g be continuous functions on I. Then their difference is also a continuous function. And since the function 'absolute value' is also continuous, the function h = |f - g| (being the composition of two continuous functions) is also continuous (on I). Now, by the so-called 'extreme value theorem' (see e.g. en.wikipedia.org/wiki/Extreme_value_theorem) the function h actually attains its sup and inf on the interval I, which justifies the use of 'max' instead of 'sup'. (the explanation above is what he calls 'takes some extra work' at 14:40) Hope this helps! (of course, if we allow a = -∞ and/or b = +∞, I is not bounded anymore and 'sup' should be used and we should restrict ourselves to bounded functions in order for the metric to even exist, i.e. not be +∞; but using square brackets in the definition of I implies that both a and b are finite)

[@iabervon]

I just noticed that Example 9 isn't even a pseudometric; if C is a blob in between A and B, d(A,B) > d(A,C) + d(C,B).

[@alegian7934]

this is kinda spooky, I've been googling metric spaces and sets in general all day today... weird that you uploaded just when I needed it

[@stevenwilson5556]

lol spooky… the simulation is stalking you.

[@xhinker]

The best video on metric space, thank you!

[@drpeyam]

Thank you!!!

[@sergioh5515]

I've been having fun watching your channel again Dr Peyam. I missed it so much! I hope you continue to make videos like these, even though I know you are a busy man! :)

[@stefanocarini8117]

Love your Crystal clear explanations!

[@Sahanie]

This is one of my favourite and most enjoyable videos from your channel. I would like to see more examples of weird metrics and topologies.

[@drpeyam]

Check out my playlist :)

[@Sahanie]

@@drpeyam Thank you, Dr. Peyem! Yes, I just noticed it in the description. I'll definitely go through it.

[@stevenwilson5556]

Never even discussed metric spaces in my undergraduate studies (BA Math, conc in Prob & Stats). Thanks for this video.

[@nailabenali7488]

Finally I've been waiting for this!! I know that you are super busy but could you make something about(Normed space in finite dimension along with arc connexity well I don't know the exact word in English but it's called connexité par arc) thank you for making math such a fun subject to learn!!

[@arturcostasteiner9735]

In English, this is called connectedness by arcs or by paths. Very similar to the French term. If a set satisfies such property, we say it's arc or path connected. Dr Peyam has made a video on this, search for Topologist Sine Curve.

[@nailabenali7488]

@@arturcostasteiner9735 Thank u !!

[@arturcostasteiner9735]

@@nailabenali7488 Vous êtes les bienvenue

[@zapazap]

Your example #9 is of a pseudometric: it possible to have d(X,Y)=0 without X=Y. A nice illustration of a metric in this context would be the Hausdorff metric (but the sets X and Y must be non-empty and compact). A nice simple illustration with practical application to strings and telecommunications would be the Hamming metric. I love your channel.

[@rembautimes8808]

I remember 30 years ago going to my local library and trying to understand a metric space but was left befuddled. Thanks to this video it’s clearer now. I guess with a lot more focus on machine learning, this will get a lot more attention. Great video , very funny too 😂

[@drpeyam]

I hope so too hahaha 😂

[@bebarshossny5148]

You make stuff fun I watch your videos to chill away from my stiff professor a bit I love a professor who enjoys teaching his subject In fact i wanna be one myself in the future

[@jamesbentonticer4706]

Are you in colleg what are you studying?

[@cediemacalisang7713]

Your way of explaining is relatable, sometimes concrete albeit always amazing.

[@IoT_]

Fun fact about Manhattan metric space : The ratio of the circumference of a circle to its diameter = 4 (Which is equal to pi in the regular euclidian metric space)

[@suayhossien]

Is that same as taxi cab

[@IoT_]

@@suayhossien yeah. It's mentioned in the video

[@thinkmachine_]

@@suayhossien Yes, it is!

[@hamzakamil185]

Awsome explaination. I just want to add that the properties of the distance are just 3 (2-3-4 in ur video), and we prouve that any distance is positve

[@channelnamechannel]

the infinity matrix norm gives a nice intuition about why iterative matrix solvers work since, for some positive definite matrix, the eigenvector with the largest eigenvalue is the stable fixed point upon iterated application of the matrix.

[@gouraviki]

Love your beautiful way of teaching...♥️

[@basedmatt]

super-clear explanation on metric spaces. Thank you very much!

[@scose]

one application of the infinity norm metric (ex. 4) is representing the time it takes to move between locations on a 2-axis machine like a milling machine or a plotter where each axis is driven independently

[@drpeyam]

Really cool!

[@willnewman9783]

Example 9 does not work. Note that if A,B have a point in common, then d(A,B)=0 even if A does not equal B. The metric on subsets I know of involves defining the distance between a point and a set first, d(a,B)= inf {d(a,b)|b in B} and then defining d(A,B)=sup{d(a,B)|a in A}

[@martinepstein9826]

That doesn't quite work either since the sup can be infinity. One way to deal with this problem is to take d(A,B) = 1 if the sup exceeds 1.

[@willnewman9783]

@@martinepstein9826 Oh yeah good point

[@adhambasheir4524]

@will newman well if A is a subset of B the your metric gives d(A,B)=0 ... so i guess we need to define d(A,B) as max{ sup{d(a,B)|a in A} , sup{d(b,A)|b in B} } @Martin Epstein about the infinity thing i guess you just can't define a number because d(A,B)

[@martinepstein9826]

@@adhambasheir4524 Good point about subsets. Dr. Peyam didn't say this in the video but a metric must map to the reals by definition, so no infinity allowed. My idea is based on the "standard bounded metric" so shouldn't introduce any problems. In your example d(A,B) = 1, not 3. The metric never exceeds 1.

[@adhambasheir4524]

@@martinepstein9826 well i guess infinity ruined every thing xD ... in my example d(a,B) = inf { d(a,b) | b in B} which is between 2 and 3 so sup{d(a,B) | a in A} = 3 .. and by symmetry d(A,B) = sup{d(a,B) | a in A} = sup{d(b,A) | b in B} = 3

[@sundaybutane9725]

If my distance function is defined as follows: d(x,x)=0, d(x,y) = c where c is a constant. Is this still a metric space? At first glance it seems to be but it feels kinda wrong.

[@alfiehellings2815]

Yes, as long as c is non zero

[@martinepstein9826]

That fits the definition as long as c > 0. You can think of this metric as less of a distance function and more of a check-for-equality function. A sequence converges in this metric iff eventually all the terms are the same.

[@suayhossien]

Yes it’s the discrete metric basically

[@suayhossien]

Basically the indicator distance lol like and on off switch, you can in fact verify it satisfies all three even triangle inequality

[@Icenri]

The case for d(infinity) maybe is that knowing that d(2) has all equidistant points in a circle and that d(infinity) has them in a square, we could assume that d(2) is obtained from x^2 + y^2 = d^2 and d(infinity) limits the exponent to infinity, in which case, max(x,y) would dominate and by taking roots d = max(x,y).

[@rolfjohansen5376]

Dear Dr.Peyam: """""WHEN"""" can I make use of matric-spaces, can assume they are 'there' when I solve a problem in real analysis? thanks

[@viktyusk]

At 3:20 the first condition is redundant: 0 = d(x, x) d(x, y) >= 0.

[@aneeshsrinivas9088]

Have you ever taken CS61B at berkeley Peyam? I actually pulled out the infinity distance during one of my CS61B projects for lighting calculations. So does this answer your question on what the infinity distance is good for.

[@drpeyam]

Thanks

[@sarojpandeya9762]

Thanks Dr Peyam

[@MathAdam]

0:24 You can hear me? Feels like I'm watching "Blink."

[@aneeshsrinivas9088]

you should make a video on what the definition of a topology is.

[@arupabinash2263]

Sir, how can I convert the theorems of point set topology into the corresponding theorems in metric spaces?I am very confused. I see there are many similarities. But, still l am unable to get a clear understanding.

[@pierreabbat6157]

A couple more metric spaces: * R^n, d, where d(a,b) is the straight-line distance from a to b if the line goes through the origin, else d(a,0)+d(0,b). * Q with the p-adic metric.

[@suleimanabubakar6569]

Which book do you recommend for the topic to me

[@suayhossien]

Could you do one on the p adic metric I’m taking an independent study on p adic numbers at the moment would like to understand them better , Damet garm

[@theproofessayist8441]

How about a metric space where we take the average of spread between two bounded sequences/functions or the minimum? Thinking inf(S). Wondering if there is applications to bigger picture theory building in other math branches.

[@drpeyam]

Minimum wouldn’t work because then you wouldn’t have d(x,y) = 0 implies x = y. Average works because it’s just the integral of |f-g| divided by the length of the interval

[@murielfang755]

Doc, could you pls recommend some texts or books on Topology? I’d like to study them during my winter vacation. Thank you! Wish u a great Christmas!

[@drpeyam]

Munkres for sure

[@murielfang755]

@@drpeyam Thank you! Will read it!

[@omara.8632]

What text book are you using to make this videos?

[@drpeyam]

Elementary Analysis by Ross

[@omara.8632]

@@drpeyam Thanks!

[@DanielL143]

A clear and enjoyable explanation. Thank-you.

[@nocomment296]

2nd video I'm watching sir

[@kaptenkrok8123]

The infinity metric is useful if you play chess. Its the distance of the kings movement on a chessboard

[@dominicellis1867]

so distance is always either the maximum distance or the minimum but its usually the maximal distance. The last one being that every distance is both a maximum and a minimum because of the forced isomorphism defined in that metric. This reminds me of the Dyson sphere in quantum mechanics that everything point on the surface is one away from the center of the sphere and in order to perform operations on a quantum object you have to work inside of the circle radiated from the surface of the sphere so as though you don't actually measure the quantum state and loose all of the work done to figure out what its going to be and why. Are these metric spaces what they mean when describing other multidimensional spacetimes like a two dimensional time axis or fractional dimensional spacetime?

[@dgrandlapinblanc]

Ok. Thank you very much.

[@azamatdevonaev1772]

Sometimes the nature of this subject makes me laugh out of nothing because I tend to laugh when I don't understand some stuff. But this "(S,d) is not San Diego" just killed me 😂

[@alejandrobarrantes8657]

thanks for your work!! I really appreciate your effort of presenting us these educational videos!!!

[@LibertyAzad]

May I ask which textbook you are using in class?

[@drpeyam]

Ross

[@Rubertoda]

Very understandable!!!

[@punditgi]

Dr Peyam reveals all!

[@jukkejukke5386]

Hello Mr Peyam. Could you please make a separate lecture about Example 8 and 9. How do I calculate the integral of the absolute of the distance of two functions? Thank you very much.

[@drpeyam]

Highly doubt it, you just use the formula in the examples

[@suayhossien]

Also the reals are a nice (complete) metric space. All Cauchy sequences go to a point in the reals aka converge

[@soumnanema8362]

Thank You!

[@briantekmen771]

The infinity metric (aka the Chebyshev distance) describes how a king moves in chess!

[@suayhossien]

TRIANGLE INEQUALITY FOR THE WIN!! What I use to prove lots of facts about Cauchy sequences lol

[@MrJapogm]

Does anybody know any good book or text for this subject?

[@drpeyam]

Yes, Ross Analysis or Munkres Topology

[@MrJapogm]

@@drpeyam Thank you! Very kind of you

[@Lawrencewalu-ru9ov]

Thank you.

[@RalphDratman]

This is brilliant

[@sagacityparagonacademy4101]

well done

[@abdifatahbarkhad8890]

Please Dr peyam talk about defferential equations one and two

[@drpeyam]

Defferential equations, the most polite out of all equations

[@revelationSandJ]

Interessantes Video 👍weiter so

[@User-gt1lu]

Can u tell me a good book for learning linear algebra (kann auch deutsch sein). ;D

[@drpeyam]

Friedberg

[@dalisabe62]

What spaces are not metric? It seems that all sets composed of real and complex numbers in any dimensional space are metric spaces. The variety of the metrics suggests a preference of one over the other based on the particular application, which I would love to explore.

[@drpeyam]

Any space is a metric space with the discrete metric

[@dalisabe62]

@@drpeyam so a discrete metric is one where the distance between any two points in the space is either zero or one. Are you saying that every metric space is necessarily a discrete metric space? My original question was: what is NOT a metric space. If I think of a space as the domain of any collection of points or objects, I would want to know how these points or objects are related to each other. It seems that every familiar space is a metric space. So are there some cases where a space is not equipped with a relation between its members? I never encountered any. I know the idea behind such formulation: is to be as abstract as possible, but there got to be some specific cases where there exists counter cases where the rule fails. I am interested in those cases so that I could appreciate the rule, if that makes any sense to you. Thanks.

[@drpeyam]

That’s what I’m saying, every set with 2 elements or more is a metric space if you put the discrete metric on it. So there are no sets that are not metric spaces. But of course you can put a distance function on a set that makes it not a metric space, such as d(x,y) = -|x-y| or d(x,y) = |x|

[@aanandimepani4824]

Metric space is T1 space?

[@MrPyromanwaterman]

You don't need "d(x,y) => 0" to be an axiom, you can prove this property of the distance with the three other axioms.

[@drpeyam]

I added it for clarity

[@aneeshsrinivas9088]

@@drpeyam question then, if a student on an exam in your class ommited proving this because this axiom is redundant and follows from the other 3 axioms. would they get points taken off?

[@skiplangly6591]

I never thought I would hear a professor make a GTA reference

[@sedenion9524]

Ty sm

[@honerzawita8024]

U are amazing

[@drpeyam]

Thank you!!!

[@MsSlash89]

Could someone please refresh my mind? Didn’t he already made a video about metric spaces? Did he just redo it better?

[@drpeyam]

No, I didn’t redo them, maybe you already looked at my playlist

[@MsSlash89]

@@drpeyam So there are two different videos introducing Metric Spaces?

[@drpeyam]

No just one, I don’t know which other one you’re referring to

[@MsSlash89]

@@drpeyam Oh so this is the first! Maybe I only dreamed about it!

[@drpeyam]

Your dream came true lol

[@lalhriatpuiahmar5057]

Sir can i know your degree

[@drpeyam]

Look at my channel name

[@valor36az]

Taxicab metric = not the shortest distance between two points

[@drpeyam]

For a taxicab it is

[@ascension7537]

This just seems like various examples of Pythagoras theorem.

[@suayhossien]

Lol a bit different there. That’s a strong equality and holds only for special cases. d is a very general version of distance between points that is defined over some arbitrary space

[@md2perpe]

People should have some self-distance. Elements in a space should not.

[@suayhossien]

We could just take the discrete metric there haha

[@aneeshsrinivas9088]

9:38 if you xould drive through the buildings, You’d think something like that would come in beta testing.

[@rikhalder5708]

What's the best book for topology? Please Dr Peyam reply me

[@kqp1998gyy]

💕

[@0x90meansnop8]

Haha social distancing metric XD

[@Neilcourtwalker]

How to kill 10 birds with one stone? Easy, just attach a stone to the blade of a windmill :-P

[@dalitsobanda7161]

Llow the jokes though that was hella cringe😭

[@kpp_]

Stop killing birds :D

[@arijitmajumder2638]

Your funny humar get math easy.