# Analysis I: eine Variable (HS22) & Analysis II: mehrere Variabeln (FS23)

In case there are any questions regarding one of the decks (“Analysis I: eine Variable (HS22)” or “Analysis II: mehrere Variabeln (FS23)”) you may ask them here. Also, if you notice a typo in one of the flashcards, I would be very grateful if you would let me know!

Do you have a deck for important proofs and theorems (like the archimedean principle) too? I’m in the fourth week of analysis I right now and your deck is very helpful, there are some errors and typos language wise but it’s no problem at all, I haven’t found an error in the important stuff until now.

Also thank you for your feedback, in case you see a more severe mistake (e.g. if the dimensions do not match, which will be relevant in Analysis II), pls let me know:)

Regarding the proofs of important theorems, etc., unfortunately I did not create an Anki deck for those, as learnt them on physical flashcards. Our professor also didn’t give us a list of important proofs (some professors do, you might want to ask prof. Figalli how he handles these things).
In case prof. Figalli doesn’t provide you with such a list, here are the ones I would recommend studying/the ones I studied, this is not an official list, so use at your own risk! (numbering is the same as in prof. Einsiedler’s script)

1. (Umgekehrte) Dreiecksungleichung (2.45 g & h)
2. Existenz des Supremums (2.59)
3. Archimedisches Prinzip (2.68)
4. Existenz von Häufungspunkten (2.75)
5. Überabzählbarkeit der reellen Zahlen (2.81)
6. Zwischenwertsatz (3.58)
7. Annahme des Maximums & Minimums (3.69)
8. Heine, gleichmässige Stetigkeit (3.75)
9. Integrierbarkeit monotoner Funktionen (4.31)
10. Riemann-Integrierbarkeit von Polynomen (4.37)
11. Konvergenz von monotonen Folgen (6.5)
12. Eigenschaften des Limes superiors (6.11)
13. Cauchy-Kriterium für Folgen (6.26)
14. Grenzwerte und Stetigkeit (6.39)
15. Leibniz-Kriterium (7.25)
16. Cauchy-Kriterium (7.26)
17. Absolute Konvergenz (7.28)
18. Gleichmässige Konvergenz und Stetigkeit (7.48)
19. Gleichmässige Konvergenz und Riemann-Intergrierbarkeit (7.49)
21. Integration von Potenzreihen (7.85)
22. Kettenregel (8.8)
23. Differenzierbarkeit der inversen Funktion (8.14)
24. Rolle (8.28)
25. Mittelwertsatz (8.29)
26. Konvexität via Steigung von Sekanten (8.39)
27. Erweiterter Mittelwertsatz (8.48)
28. Ableitung des Integrals (9.2)
29. Taylor-Approximation (9.46)

As I said, this is not an official list, however, in our exam both proofs that were asked are contained in the above list (7.48 & 4.31).

In general, when it comes to selecting proofs to study, I’d say it’s important to focus on the thms. etc. that provide you with important and often used results. Furthermore, the ones that are not extremely long/short and the ones that are not too technical could also be good candidates for exam questions.

Also, when studying proofs i recommend writing down a few bullets points (to learn by heart) with important steps for each proof and then trying to fill the missing parts yourself instead of learning every single word, equation, etc. by heart. That way you also get more confident writing your own proofs and there’s less of a risk of having a black out during the exam.

Hope that helps! PS: don’t worry about exams too much, it’s way too early for that in the semester;)

Wow, I didn’t expect you to answer that quick and in so much detail ^^ .
Thank you very much for the many great points you gave and the list, I’ll note both down somewhere (and will also ask Figalli about the proof list thing). One other question: Did you make the physical flashcards for your proofs yourself?
Also: When did you start studying for the Basisprüfungsblock 1 exams? I’m currently a little overwhelmed with the workload and I don’t know where I should fit all the practice from books (e.g. Michaels) in that I’d need to actually get what’s going on in these exercises (and I’m not so sure about it fitting in the one month of preparation for the exams).
May I also ask you in which semester you are and what you’re studying?

Question to a card in your Analysis I deck, category 2 Reelle Zahlen:
I quote: "(\text{The set of real numbers: }) {{c1::(\text{A set with } \quad +;:; \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \quad \cdot;:; \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \quad \text{and } \leq \text{ on } \mathbb{R} \text{ such that the 16 field axioms hold.};)}}
My question is now, 16 field axioms? 1. I thought completeness is an additional axiom for the real numbers since it isn’t in the field axioms. 2. Shouldn’t it be “… ordered field axioms”, since you included the ordering?
3. Just that I’m not making a big oopsie, a field is basically two abelian groups with +,* and an additional property, that is, the distributive property, right? So only 10

No problem!

Yes, I made the proof flashcards myself, not during the semester though, but during the study phase. I spent about 20-30min on each proof when writing the flashcards (to understand the proof itself and trying to “summarise” it in such a way that I’d be able to recall it).

Depends on what you mean with “studying for the BP1”, during the semester I focused on understanding the material and solving the weekly exercise sheets. I really recommend solving those! In case I don’t know how to approach a certain problem I usually first go back to the script/lecture notes to see if there’s something I could use (e.g. when you get to power series there are several theorems that might come in handy when calculating the convergence radius of a specific series), if I’m still not successful my next step is to ask friends, and then, if still necessary, ask a TA (e.g. in the study center).
Back then, my TA told us that we should spend about 2h per exercise on average (in Analysis).
That being said, I didn’t start solving exc. from other literature (e.g. Michaels) until the preparation time after the semester started. Generally, the exercises they provide you with are rather good ones (there are exceptions…), so I suggest focusing on those during the semester and understanding essence of the theorems, propositions, lemmata, etc.

I’m a mathematics undergrad, currently in my third semester:)

You’re right, this flashcard isn’t correct. I’ll definitely update it, thx for pointing it out!
In case you you don’t want to download the complete deck again, here’s the updated version for you to copy-paste:

(\text{The set of real numbers: }) {{c1::(\text{The set $(\mathbb{R}, \leq)$ together with } +;:; \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, ;\cdot;:; \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \text{ such that $(\mathbb{R}, \leq)$ is a complete, ordered field, i.e. the 16 axioms of a complete, ordered field hold.} )}}

Regarding the questions:

1. Yes, the completeness axiom is an additional axiom.
2. I Corrected that as well
3. Almost, consider a set K s.t. (K,+,e_1) and (K,\cdot , e_2) are (different) abelian groups where e_1 and e_2 are the neutral elements. For (K,+,\cdot ,e_1,e_2) to be a field, it must satisfy two more axioms, one being (left-)distributivity (right-distributivity follows from commutativity & associativity), as you correctly mentioned. This, however, only makes 9 axioms and does not define a field yet. An important one that many people forget in the beginning is non-triviality, this means that the two neutral elements cannot be the same, i.e. e_1 != e_2. (in other words, a field must contain at least two elements, the neutral elements).

In case you haven’t seen the field F_2 yet, you might want to check it out, it’s the field with only two elements:)

Also, if you’re a mathematician like me, you’ll see fields again next semester in the course “basic structures” (where you will even construct the real numbers using for example Dedekind cuts or Cauchy-sequences), and then at least one more time in Algebra I together with groups and rings. In those later courses you will go into more depth;)

I see, I did not once go back to the script/lecture notes when solving the exercises up until now (always directly asked friends and they mostly explained the whole thing to me), that’s probably one of the reasons I only understand about 20% of the exercises .-. Good tip! I’ll try to focus more on the exercises and the theorems and stuff, I guess I was stressing myself too much with thinking I have to help myself with other literature and work through them.
I think 20-30min per proof is pretty fast, damn. I probably could spend hours on one I’m a physics undergrad btw so I won’t have the Grundstrukturen course (I think). Thanks for the answer and corrections and I have another question: What is the F_2 field good for? The two elements have to be {0,1} respectively the neutral element for addition and multiplication right? What can you do with such a field?
PS: 1. In the “Basisjahr::Analysis I: eine Variable::2. Die reellen Zahlen” deck, the card for Subtrakion in N is partly false (I think). Shouldn’t the < in (\forall n,m \in \mathbb{N}, ; m < n : m-n \in \mathbb{N}) be a >, since otherwise m-n would be negative and not inside \mathbb{N}?
2. What does the \sqcup stand for in “\mathbb{Z} = \mathbb{N} \sqcup {0} \sqcup {-n | n \in \mathbb{N}} = \mathbb{N}_0 \sqcup -\mathbb{N}” ? Is it like in … union with … ?
3. Regarding the complete, ordered field axioms: 2 different abelian groups + distributivity makes 9 axioms as you said, then with non-triviality it’s 10… what are the remaining 6 axioms then? Are they the axioms of the order relation: reflexive, anti-symmetric, symmetric and transitive? If yes then there still remains 2?

Other literatur might provide you with good exc., which for example Michaels focuses on, but the material provided by prof. Figalli should suffice when it comes to understanding the material - if for some reason his lecture notes/script aren’t well-structured I can really recommend having a look at prof. Einsiedler’s script from the past two years, I know it’s a rather long one (800 pages) but I think it’s really good! (we covered chapters 1-9 in Analysis I and the remaining chapters in Analysis II)

That’s the time I spent writing a flashcard for each proof, I obviously spent way more time actually studying them In that case no, you won’t have Grundstrukturen, but as far as I’m concerned you can choose it if that’s something you’re interested in. As a fair warning tho, it’s quite a bit more abstract than Analysis & LinAlg, e.g. you’d have axiomatic set theory, which I personally struggled with a lot in the beginning, for me it just took more time to digest the material and make sense of it compared to the other subjects. On the other hand, you’d get to see some cool things like the axiom of choice:)

F_2 is one of the standard examples for finite fields, in fact, F_p is a field for every p prime (if you’re unsure about the definition of a field, it would be a good exercise to show that F_p is a field, to make it easier you might choose a specific p prime), careful tho, the F_p’s aren’t the only finite fields! (This is part of Algebra II)
You may think of the “group operations” as addition and multiplication, however, in a more general sense this isn’t really accurate, the two “group operations” can be any binary operation s.t. it forms a group with the given set. The same goes for 0 and 1.
When it comes to applications of finite fields, I’m honestly not so sure, as I said, those will be more thoroughly discussed in Algebra II, which I will probably take next semester, but even there we probably won’t see actual applications, we’re mathematicians, we don’t really care about real world applications, that’s your job as a physicist and other people’s jobs But from what I’ve heard so far, finite fields are used in number theory, cryptography, and some other parts of computer science, since a computer can only represent numbers with finitely many digits (you’ll also have an intro to floating point numbers in your computer science course if you haven’t had it already, but you won’t discuss finite fields there).

1. yes, that’s a typo
2. \sqcup is the disjoint union of two set, so if X and Y are set s.t. X \cap Y = \emptyset and we want to write X \cup Y (under the previously mentioned assumption), then we can use \sqcup as a short-hand the notation. Some lecturers also write a dot above \cup to indicate disjoint union.
3. The axioms are the ones from the abelian groups, distributivity, then reflexivity, transitivity, antisymmetry, and linearity of the ordering relation. Then we require the ordering relation to be compatible with both group operations, and lastly the completeness axiom. (Here we didn’t specifially mention the non-triviality, this kind of depends on the definition of a complete, ordered field you were provided, it might have been introduced to you a bit differently, we just said that non-triviality can be deduced from the above axioms which is why we didn’t mention it seperately anymore, but it is needed if you want to show that something is a field).
I also made a flashcard for each those axioms in case you haven’t seen them yet, and I wrote them down in the “extra” section in the card you mentioned previously as well.
1 Like

Alright, Thanks a lot!
So Symmetry isn’t an axiom of the complete, ordered field even though it’s in the relation axioms?

No, I think you’re mixing up the definitions of equivalence relation and ordering relation:

An equivalence relation must satisfy reflexivity, symmetry, and transitivity.

An ordering relation, however, must satisfy reflexivity, antisymmetry, and transitivity. If it also fulfills totality, i.e. for all x,y we have x <= y or y <= x, we say <= defines a total ordering. Then there is also the term well-ordering which requires one more thing, which I won’t go into detail hear.

If we asked for symmetry instead of antisymmetry the following would have to hold: for all x,y : x <= y ==> y <= x ; a specific example would be: 1 <= 2 ==> 2 <=1
Which doesn’t really make sense, does it? 1 Like

Die Reihe muss doch nicht automatisch eine Nullfolge sein, bzw. es gilt doch nicht für alle Reihen, dass sie wenn sie diese Form haben und konvergieren, eine Nullfolge sind, oder doch?

Eine weitere Verständnisfrage hier: Müsste S hier nicht nur eine Basis sein, sondern das Erzeugendensystem bzw. die lineare Hülle (also mehrere)? Es kann ja nicht jeder Vektor aus V als Linearkombination von einer Basis geschrieben werden, dafür braucht es doch mehrere Basisvektoren, die dann die lineare Hülle bzw. Span (oder halt Erzeugendensystem) bilden?

Bzgl. der Analysis Frage:

Mit Nullfolge ist hier eine Folge, die gegen Nullkonvergiert gemeint, nicht die folge welche konstant Null ist.

Der Beweis für diese Aussage ist ziemlich einfach:

Per Annahme konvergiert die Summe gegen einen Grenzwert, sei dieser S. Dies bedeutet in anderen Worten, dass die Folge s_n = \sum_{k=1}^n a_n gegen S konvergiert. Betrachte nun den Limes von s_n - s_{n+1} = a,{n+1}. Per Annahme konvergieren s_n und s{n+1} gegen S, woraus wiederum folgt, dass \lim_{n \to \infty} a_n = \lim_{n \to infty} (s_n - s_{n+1}) = S - S = 0, also konvergiert a_n gegen 0.

Ich hoffe das hat die Frage geklärt.

Bzgl. LinAlg:

Die Definition einer Basis lautet wie folgt:

Eine Teilmenge S \subseteq V ist eine Basis von V, falls folgende zwei Bedingungen erfüllt sind:

1. S erzeugt V
2. S ist eine linear unabhänige Menge

D.h. Basis impliziert Erzeugendensystem.
Ein wichtiger Unterschied hierbei, ist dass ein einzelner Basisvektor nicht zwingenderweise eine Basis bildet (abgesehen vom Fall, dass V 1-dimensional ist). Eine Basis hingegen ist eine Menge von Basisvektoren (d.h. l.u.), die den Raum aufspannen (siehe Def.)

Bem.: Ein Basisvektor kann nicht der Nullvektor sein

Falls dich dies interessiert wäre hier noch der Beweis der Proposition:

Bemerke, dass wir die Annahme, dass V endlich dimensional ist treffen. In LinAlg I & II werden hauptsächlich endlich dimensionale Vektorräume betrachtet (es gibt Aussnahmen!). Wenn ich mich nicht irre sind unendlich dimensionale Vektorräume Teil der Vorlesung Funktionalanalysis (ein Kernfach der reinen Mathematik).

1 Like

Müsste es nicht bei monoton und streng monoton fallend für alle x,y : x < y sein? Und auch allgemein bei den anderen statt kleiner gleich direkt <?
Habe hier auch nochmal in unserem Skript nachgeschaut, da steht es auch so:

Danke für die anderen Antworten, ich gehe die bald durch ^^

jap, ist ein Typo, thx