Functional Analysis 5 | Cauchy Sequences and Complete Spaces

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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
02:30 Cauchy sequences
03:34 Complete metric spaces

#FunctionalAnalysis
#VectorSpaces
#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)

[angelmendez-rivera]

Do not be confused, people! (0, 3) is not a closed set of the metric space (R, d), but it is a closed set of the metric space ((0, 3), d), which is what the video claimed. Also, that there are Cauchy sequences that do not converge in ((0, 3), d) does not mean (0, 3) is not closed in ((0, 3), d), but that ((0, 3), d) is not complete. This is the gist of what the video is trying to say.

[zazinjozaza]

These videos are amazing for building intuition.

[cediemacalisang]

You don't know how thankful I am.

[prank]

Alles super erklärt. Ich freue mich jetzt schon auf die nächsten Videos!

[Independent_Man]

Take any closed subset A of a complete metric space. Then, the set A with the same inherited metric is a complete metric space in its own right.

[PunmasterSTP]

Complete metric spaces? More like "All of these concepts put us through our paces!" Though it can be difficult to grapple with these ideas, I think you're doing a wonderful job of shepherding us through them. Thank you!

[qiaohuizhou]

2:30 completeness
3:15 the Cauchy sequence is always a generalization of convergent sequence
4:50 Example of a complete metric space with discrete metric but is incomplete with Euclidean metric

[umbranocturna]

Im not writing Analysis this semester, but looking forward to learn with this next semester.

[tlli]

Dr. Großmann is there any textbook you would like to recommend, for which contains exercises which corresponds to your lectures.

[xiangqerupiso]

would you be so kind and make a videoseries about stochastiks?
would really appreciate it

[javohirsultonov]

Why is the set only closed? We proofed in part 4 that set is closed when all of its sequences converge to x belongs this set. Here you showed example of sequence that doesn't satisfy it, I mean (1/n), then why is it still closed?

[Afxonidis]

I have a question. The idea of having a hole in the space X, also extends to sequences that doesn't converge on R^3 or on R^2 spaces? Should I think it as a real hole eg on the plane (xy) for R^2?

[jordanfernandes]

Hi. Thank you for sharing these videos. I'm sort of confused about whether the sequence summing terms 1/n over the positive natural numbers (is this the harmonic series ?) converges ?! Intuitively it feels like it should right because the terms in the denominator grow to inifinity but I've been reading that it does not converge. Is that true and if yes, is there an easy way to see why ?!

[ouafaeraibi]

Hello, first 1000 thanks for the videos they are all amazinly explained
my question is about the set (0, 3) being closed . if we consider the sequence 3-1/n then all its elements belong to X but still its limit (=3) doesn't . ?? so why the set is closed ? please correct me if I am wrong

[ichkaodko]

I am wondering why some of your video has no english subtitle? do you make it or is it automatic?

[trondsaue]

In the video we learn that X=(0, 3) with the usual metric is closed, and that one argument for this is that the complement the empty set is open. Now in lecture 3 a definition of open sets were given in terms of epsilon-balls. Do the empty set comply with that definition ?

[xwyl]

Normally, the name Cauchy always gives me a headache. But this time at The Bright Side, it isn’t the case.

[oskarjung]

6:39 How is b) a complete metric space? Because, not all cauchy sequence converge, as defined above.
In this case, if epsilon is >1 then there are so many violent sequences that will be cauchy sequences, because d(x(n), x(m)) is always <=1 for all possible x(n), x(m). So, even divergent sequences will become Cauchy sequences. Hence, the condition of all Cauchy sequence converge is not satisfied. Hence, it should not be a complete space.

[ZirTaaah]

:) I Think I miss something :p sequence 1/n is in X but not his limit so X is not closed or we cant apply sequential characterization cuz we dont have cauchy Space ??

[rusumbetov]

If limit is 0 then we can say that the sequence is not convergent?