What is Real analysis: Definition and 501 Discussions

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.

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  1. L

    I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?

    Hi everyone in the following expression ##f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega ## the book says I can't swap integrals bacause the function ##f(u) e^{i \omega(t-u)}## is not ## L^1(\mathbb{R} \times \mathbb{R})## why ? complex...
  2. S

    On the ratio test for power series

    In these lecture notes, there is the following theorem and proof: I'm confused about "...the power series converges if ##0\leq r<1##, or ##|x-c|<R##...". In other words, why is ##|x-c|<R## equivalent to ##0\leq r<1##? I guess the author reasons as follows. If $$R=\lim _{n\to \infty...
  3. I

    Prove ##(a+b) + c = a + (b+c)## using Peano postulates

    I have to prove the associative law for addition ##(a+b) + c = a + (b+c)## using Peano postulates, given that ##a, b, c \in \mathbb{N}##. Now define the set $$ G = \{ z \in \mathbb{N} |\forall\; x, y \in \mathbb{N} \quad (x + y) + z = x + (y + z) \} $$ Obviously, ## G \subseteq \mathbb{N} ##...
  4. caffeinemachine

    I Feedback for my YouTube Videos on Real Analysis

    Some time back I posted about my videos on Group Theory on YouTube and got valuable feedback from the PF community. With the response in mind, I made substantial changes to my presentation. One of the main complaints was that I was speaking too fast. Here is my recent video on Real Analysis...
  5. bhobba

    B What Are Infinitesimals?

    When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...
    • bhobba
    • Thread
    • Calculus infinitesimals Real analysis
    • Replies: 24
    • Forum: Calculus
  6. L

    Finding the measure of a set

    my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}
  7. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  8. J

    Proof for Real Analysis

    Proof: Suppose f is a function and x is in the domain of f s.t. there is a derivative at the point x and sppse. there are two tangent lines at the point (x,f(x)). Let t1 represent one of the tangent lines at (x,f(x)) and let t2 represent the other tangent line at (x,f(x)) s.t. the slopes of t1...
  9. C

    Problems with Real Analysis class

    Hi. I'm nearing retirement so I thought I would take some math classes. This fall I took a Real Analysis class at a good school and dropped it because I did so bad on the first exam. I did great on the homework and quizzes. I also took Real Analysis about 47 years ago at a very good school...
  10. MexChemE

    Analysis Study plan for Functional Analysis - Recommendations and critique

    Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
  11. H

    Proving that there is a sequence in S, such that ##\lim s_n = \sup S##

    Let ##S=\{s_n:n∈N\}##. ##\sup S## is the least upper bound of S. For any ϵ>0, we have an m such that ##\sup S−\epsilon \lt s_m## ##\sup S−s_m \lt \varepsilon## ##|\sup S−s_m| \lt \varepsilon## I mean to say that, no matter how small ϵ is, there is always an element of S whose distance from supS...
  12. I

    If ##x> 1## and ##x^2 <2##, prove ##x < y##, ##y^2<2##

    Suppose ##x \in \mathbb{Q}## and ##x > 1## and ## x^2 < 2##. I need to come up with some ##y \in \mathbb{Q}## such that ##x < y## and ## y^2 < 2##. Here is my attempt. Give that ##x > 1## and ## x^2 < 2##, I have ## (2-x^2) > 0## and ##4x > 0##. Also, ##2x >0##. Now define $$ \alpha = \text{...
  13. H

    Proving that an Integer lies between x and y using Set Theory

    ## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...
  14. H

    Proving a property of a Dedekind cut

    A Dedekind cut is a pair ##(A,B)##, where ##A## and ##B## are both subsets of rationals. This pair has to satisfy the following properties A is nonempty B is nonempty If ##a\in A## and ##c \lt a## then ##c \in A## If ##b \in B## and ## c\gt b## then ##c \in B## If ##b \not\in B## and ## a\lt...
  15. C

    Which of the following statements are true? (Real Analysis question)

    Summary:: x Problem: Let ##f:[0, \infty) \rightarrow \mathbb{R}## be a positive function s.t. for all ## M > 0 ## it occurs that ## f ## is integrable on ## [0,M] ##. Which of the following statements are true? A. If ##\lim _{x \rightarrow+\infty} f(x)=0## then ##\int_{0}^{\infty} f(x) d x##...
  16. PseudoQuantum

    Analysis Real Analysis Preparation

    Hi all. I am a math major. I will be taking real analysis next Fall with an excellent professor who I know to be also quite demanding. I would like to be as well prepared for this class as possible besides going through a real analysis text or lecture series over the Summer and causing the class...
  17. Math Amateur

    MHB Exploring Proposition 6.1.2 from D&K's Multidimensional Real Analysis II (Integration)

    I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...
  18. C

    I In Euclidian space, closed ball is equal to closure of open ball

    Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...
  19. docnet

    Real analysis: prove the limit exists

    Prove that each of the limits exists or does not exist. 1. ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## if ##\forall \epsilon>0, \exists \delta ## such that ##|x-2|<\delta \Rightarrow |f(x)-3|<\epsilon##. \begin{align}&|x^2-1|=|x+1||x-1|\leq \epsilon\\...
  20. H

    Prove that the inner product converges

    I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...
  21. C

    I What's the definition of "periodic extension of a function"?

    I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...
  22. C

    Prove limit comparison test for Integrals

    Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
  23. U

    I Limit of limits of linear combinations of indicator functions

    I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
    • Unconscious
    • Thread
    • Combinations Functions Limit Limits Linear Linear combinations Measure theory Real analysis
    • Replies: 3
    • Forum: Calculus
  24. Mr.Husky

    Other Learning Real Analysis at My Own Pace

    Hi everyone, I recently started studying real analysis from baby Bruckner couple. It feels me like, "I am running too fast to reach my destination but in the process of running, I decreased my oxygen level." So, I stopped trying to complete uni coursework fast. But rather I started reading...
  25. Mr.Husky

    Analysis Opinions on textbooks on Analysis

    What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them. Note- I am now studying real analysis from...
  26. C

    Showing continuous function has min or max using Cauchy limit def.

    Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
  27. C

    I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

    Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
    • CGandC
    • Thread
    • Limit Limits Partial Real analysis Sequence and series Subsequence
    • Replies: 13
    • Forum: Topology and Analysis
  28. Falgun

    Analysis Real Analysis (Baby Rudin vs Apostol)

    I am currently trying to self study Real analysis . I have completed Hubbard's Multivariable book & Strang's Linear algebra book. I have Apostol's Mathematical Analysis & Baby Rudin . I started with rudin yesterday and was making excellent headway until I encountered a theorem about 15 pages in...
  29. yucheng

    Can I use recursion/induction to show that N <= x < N+1 for x real

    Homework Statement:: Show that for every real number ##x## there is exactly one integer ##N## such that ##N \leq x < N+1##. (This integer is called the integer part of ##x##, and is sometimes denoted ##N = \lfloor x\rfloor##.) Relevant Equations:: N/A I have tried reading the solution given...
  30. yucheng

    Is my proof that multiplication is well-defined for reals correct?

    I have referred to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ to check my answer. The way I thought of the problem: I know ##xy = \mathrm{LIM}_{n\to\infty} a_n b_n## and I know ##x'y = \mathrm{LIM}_{n\to\infty} a'_n b_n##. Thus if ##xy=x'y##, maybe I can try showing...
  31. yucheng

    Understanding the Use of Min in Cauchy Sequences

    I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/ I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
  32. yucheng

    I Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady

    In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
  33. yucheng

    Proof that two equivalent sequences are both Cauchy sequences

    Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##. A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##. We proceeded by...
  34. S

    B Is complex analysis really much easier than real analysis?

    This author seems to say so: https://blogs.scientificamerican.com/roots-of-unity/one-weird-trick-to-make-calculus-more-beautiful/
    • swampwiz
    • Thread
    • Analysis Complex Complex analysis Real analysis
    • Replies: 4
    • Forum: Calculus
  35. Adesh

    I Will ##M_i = m_i## if an interval is made vanishingly small?

    We define : $$M_i = sup \{f(x) : x \in [x_{i-1}, x_i ] \}$$ $$m_i = inf \{f(x) : x \in [ x_{i-1}, x_i ] \}$$ Now, if we make the length of the interval ##[x_{i-1}, x_i]## vanishingly small, then would we have ##M_i = m_i##? I have reasons for believing so because as the size of the interval is...
    • Adesh
    • Thread
    • Interval Real analysis
    • Replies: 14
    • Forum: Calculus
  36. Adesh

    I How to prove that ##f## is integrable given that ##g## is integrable?

    We have a function ##f: [a,b] \mapsto \mathbb R## (correct me if I'm wrong but the range ##\mathbb R## implies that ##f## is bounded). We have a partition ##P= \{x_0, x_1 , x_2 \cdots x_n \}## such that for any open interval ##(x_{i-1}, x_i)## we have $$ f(x) =g(x) $$ (##g:[a,b] \mapsto \mathbb...
    • Adesh
    • Thread
    • Integrability Real analysis
    • Replies: 30
    • Forum: Calculus
  37. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
    • Adesh
    • Thread
    • Calculus Function Integrability Real analysis Sums Upper bound
    • Replies: 5
    • Forum: Calculus
  38. CaptainAmerica17

    I Help Finding the Correct Approach to this Proof (Intro Real Analysis)

    Ok, so here is what I have so far: Suppose ##T_1## is infinite and ##\varphi : T_1 \rightarrow T_2## is a bijection. Reasoning: I'm thinking I would then show that there is a bijection, which would be a contradiction since an infinite set couldn't possibly have a one-to-one correspondence...
  39. CaptainAmerica17

    Would someone mind checking my proof? Intro Real Analysis

    Here is my solution. I used mathjax to type it up in Overleaf. I feel like it makes sense, but I also have a feeling I might have "jumped the gun" with my logic. If it is correct, I would appreciate feedback on how to improve it. Thanks!
  40. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  41. C

    I Finite expansion of a fraction of functions

    I am having a problem finding the right order above and below to find the finite expansion of a fraction of usual functions assembled in complicated ways. For instance, a question asked to find the limit as x approaches 0 for the following function I know that to solve it we must first find...
    • Cha0t1c
    • Thread
    • Expansion Finite Fraction Functions Real analysis
    • Replies: 11
    • Forum: Calculus
  42. CoffeeNerd999

    I Do I need induction to prove that this sequence is monotonic?

    I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$ y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$ x_{j} \leq y_n.$$...
    • CoffeeNerd999
    • Thread
    • Induction Real analysis Sequence Sequence and series
    • Replies: 1
    • Forum: Calculus
  43. zeronem

    Introductory Real Analysis Problem

    $$r<x<s$$ $$s-r>0$$ We enploy the Archimedean principle where $$n(s-r)>1$$ We employ density of rationals where $$\exists [m,m+1] \in Q$$ Such that $$nr\in [m,m+1)$$ Therefore $$m\leq nr \lt m+1$$ $$ \frac m n \leq r \lt \frac m n + \frac 1 n $$ Since $$ \frac m n \leq r $$ Then $$...
  44. AlmX

    Analysis Is Baby Rudin a good choice for first my Real Analysis textbook?

    Summary: Is Baby Rudin a good choice for first Real Analysis textbook for someone without strong pure math background? I've completed 2 semesters of college calculus, but not "pure math" calculus which is taught to math students. I'm looking for introductory text on Real Analysis and I've...
  45. D

    I Rudin Theorem 1.21: Maximizing t Value

    Summary: Rudin theorem 1.21 He has said that as t=X/(X+1) then t^n<t<1 then maximum value of t is 1. then in the next part he has given that t^n<t<x. as maximum value of t is less than 1 why has he given that t<x ?
    • Darshan
    • Thread
    • Real analysis Theorem
    • Replies: 6
    • Forum: Calculus
  46. P

    Help with a real analysis problem

    I tried to prove this by absurd stating that there is no such ## \mu'## but i couldn't get anywhere...
  47. MidgetDwarf

    Intro Real Analysis: Closed and Open sets Of R. Help with Problem

    For the set A: Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## . By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...
  48. Tonzzi

    Real Analysis Textbook Recommendation

    Does anyone have a recommendation for a book(s) to use for the self-study of real analysis? I have just finished Apostol Calculus, Vol. 2 and would like to move on to real analysis. I am not sure whether I should continue following Apostol and move on to Apostol mathematical analysis or...
  49. V

    A What type of function satisfy a type of growth condition?

    Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
    • Vulture1991
    • Thread
    • Condition Function Functional analysis Growth Real analysis Type
    • Replies: 2
    • Forum: General Math
  50. anhtudo

    I Lemma 1.2.3 - Ethan.D.Bloch - The Real Numbers and Real Analysis

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