Functional Analysis 3 | Open and Closed Sets

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This is my video series about Functional Analysis where we start with metric spaces, talk about operators and spectral theory, and end with the famous Spectral Theorem. I hope that it will help everyone who wants to learn about it.

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00:00 Introduction
00:37 Epsilon ball
01:35 Notions
06:24 Examples

#FunctionalAnalysis
#Mathematics
#LearnMath
#calculus

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[debforit]

The example of openness, closeness and boundaries towards the end of the video is an absolute gem... shows that blind intuition (drawing analogies mainly from the continuous realm of reals) can deceive us more often than not...

[martinschulze]

For all the people who are confused why in the last example (1, 3] is open.... thats because the epsilon-ball B(x) = { y element X | d(x, y) < eps) and when you consider the point at x=3, there are no points complementary to A between 3 and 4, is just not part of the metric space X by definition, so the eps-ball definition gives you a thumbs up and this is why its an open set

[mmarques]

I'm having fun with these short lectures! I'm super curious to see how far this will go into the topic. Congratulations on your ability to explain these concepts so clearly!

[PunmasterSTP]

Dang that example with X := (1, 3]U(4, ∞) and A := (1, 3] really threw me for a loop. If I understand correctly, the point x = 3 is *not* a boundary point because its epsilon-ball when d < 1 lies entirely in the region that is less than 3, since the region (3, 4] doesn't exist in the set X. Hence the epsilon-ball is entirely contained in A, showing that x = 3 isn't on a boundary but rather is in the interior of A. Now that I (think I) understand that concept, it is *very* cool!

Man, these videos are ballin!

[cricanalysis]

These concepts are a nightmare when self studying. You made them a little easier to understand. Thanks man !

[ekaptsv]

The last example of closure was very enlightening to me! Thank you and, please, continue doing examples!

[phyllis]

I'm a Korean student, and recently started studying college level mathematics for double-major. This is the part I've been having a hard time with these days, and this video was very helpful. Your other videos are very cool, too. Thanks alot! :D

[qiaohuizhou]

0:30 if you fix a point x in a matrix space, you can look all the other points that have the same distance from the point x
0:44 the notion of the ball we want to generalize in a metric space
1:20 the epsilon ball is never empty
1:40 open set
3:15 boundary point
4:58 open set A is the set where all the boundary points of A do not belong to A
5:10 closed set define using the above logic
5:29 the definition of closed set
5:52 closure
6:33 example : X is a subset of real number
6:55 considering subset of X

[eamon_concannon]

6:30 In the example, A is a subset of X and 3 is an element of A, so 3 is either a boundary point or not of A. If 3 is a boundary point of A then any open ball centered at 3 will contain elements of A and elements of X/A. In particular, a ball of radius 1 centered at 3 will only contain A and (3, 4]. Since (3, 4] is not in X/A = (4, infinity), then 3 is not a boundary point of A.
Since none of the points of A are boundary points of A, we can say that A is open.
Also, since the complement of A is open, then A is closed.
I think that you could use a ball centered at 3 with any radius < 1 and come to the same conclusion.
I think the confusion was in considering the complement of A not as X/A = (4, infinity) but as all real numbers outside of (3, 4]

[wenanyaugustine]

Am really enjoying this lecture. I can't believe after 5 years I can still enjoy maths like this.The professor i very clear, he is more of a teacher than a lecturer.

[VietNguyen-vifu]

I love your lectures on measure theory and analysis. This also helps me to understand the concepts clearer :-)

[sinanakhostin]

The important subtlety in the last example is that A would not be an open substet X if X were to be the whole real line R. It is sometimes confusing because we tend to forget that our set X actually has a hole in it. in another word {3< x <=4} dose not exist in X. therefore the open epsilon ball B_epsilon(3) only considers the values less than 1 inside (2, 3]. In other word the interval (3, 4] is a void.

[mikhail_kochubey]

I am a Russian student and I found your videos really helpful and useful. Thank you !

[wenzhang]

The last example is very helpful, thank you

[jhkeum]

Could I ask a question?
On Example (a), you have proven that 'For x=3' B_1(x) belongs to A and that A is open.
Epsilon was assumed(?) 1, so it turned out that it doesn't contain 4 in set X.
How does it make sense that the result from assumed epsilon could define the general characteristic of a subset A?

+)
I could not understand that openness and closedness is not an opposite pair of notion. Into which video could I check?

[lucasfreitag]

I’m really excited to see more functional analysis videos. I have an exam coming and you’re really helping me out. Could you maybe make a video about operators (why boundedness implies continuity, Neumann series and investable operators). Also something about properties of infinite dimensional spaces like the theorem of Arzelà-Ascoli. The most difficult topic, so I find, is however the Sobolev spaces (although it might be too early). You’re a great teacher so please continue with this playlist haha.

[annali]

this is super helpful, really neat video and clear explaination

[mctab]

Thank you for these lectures! I have a question about last example (a): if we consider X:=R instead of X presented in the lecture then A won't be open, right?

[MrWater]

Incredible well explained! Love it! I'm thinking about what you said, points in A that are boundary points but are inside A, I always think of boundary points as points that are not inside A. I'm going to think about it

[samhughes]

I'm a tiny bit confused by the last statement. Wouldn't C_bar equal (1, 2) because 2 has been shown to be a boundary and not included in (1, 2]. The compliment of C would be (2, 3], (4, inf)?