Math 55103
Theory of Functions of a Real Variable II
Spring 2025

Information on this syllabus is subject to change. Any updates will be posted to this site. In particular, any required changes in course delivery, such as transition to remote delivery only, may necessitate significant modifications to this syllabus.

Homework assignments and important dates

List of lecture problems: PDF, HTML

	Date	Event
	Monday, Jan. 13	First day of class
	Friday, Jan. 17	Last day to add a course
	Monday, Jan. 20	Martin Luther King day (no class)
	Monday, Jan. 27	Last day to drop a course
HW 16	Sunday, Jan. 26, 2025, 11:59 p.m.	(HW 16.1) Let \(1\leq p\leq\infty\) and let \(q\) satisfy \(1/p+1/q=1\). Let \(E\subseteq\mathbb{R}\) be measurable. Suppose that \(f\in L^p(E)\) and that \(\int_E fg=0\) for all \(g\in L^q(E)\). Show that \(f=0\) almost everywhere in \(E\). Suppose that \(f\in L^p(E)\), that \(S\subset L^q(E)\) is dense, and that \(\int_E fg=0\) for all \(g\in S\). Show that \(f=0\) almost everywhere in \(E\). (HW 16.2) Let \(E\subseteq\mathbb{R}\) be measurable and suppose that \(m(E)\gt0\). Show that \(L^\infty(E)\) is not separable. (HW 16.3) Prove an analogue to Proposition 8.2 for sequence spaces. That is, recall that \(\ell^p\) is the space of all sequences of real numbers \(\{a_n\}_{n=1}^\infty\) such that the \(\ell^p\)-norm \(\bigl\\|\{a_n\}_{n=1}^\infty\bigr\\|_{\ell^p}=\Bigl(\sum_{n=1}^\infty \|a_n\|^p\Bigr)^{1/p}\) or \(\bigl\|\{a_n\}_{n=1}^\infty\bigr\\|_{\ell^\infty}=\sup_{n\in\mathbb{N}} \|a_n\|\) is finite. Let \(1\leq p\leq \infty\) and let \(q\) be the conjugate exponent, so \(1/p+1/q=1\) (with \(q=1\) if \(p=\infty\) and vice versa). Let \(\{b_n\}_{n=1}^\infty\in \ell^q\) and define \(T:\ell^p\to\mathbb{R}\) by \(T(\{a_n\}_{n=1}^\infty)=\sum_{n=1}^\infty a_n\,b_n\). Show that the sum is convergent for all \(\{a_n\}_{n=1}^\infty\in \ell^p\) (and so \(T:\ell^p\to\mathbb{R}\)), that \(T\) is a bounded linear operator, and that \(\\|T\\|_*=\\|\{b_n\}_{n=1}^\infty\\|_{\ell^q}\). Be sure to include all of the three cases \(p=1\), \(p=\infty\), and \(1\lt p\lt \infty\). A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 17	Sunday, Feb. 2, 2025, 11:59 p.m.	(HW 17.1) Prove an analogue to Lemma 8.4 for sequence spaces. That is, suppose that \(1\leq p\leq\infty\), that \(0\leq M\lt\infty\), and that \(\{b_n\}_{n=1}^\infty\) is a sequence of real numbers such that, if \(\{a_n\}_{n=1}^\infty\in \ell^p\), then \(\sup_{k\in\mathbb{N}} \bigl\|\sum_{n=1}^k a_n\,b_n\bigr\|\leq M\\|\{a_n\}_{n=1}^\infty\\|_{\ell^p}\). Show that \(\{b_n\}_{n=1}^\infty\in \ell^q\), where \(1/p+1/q=1\), and that \(\bigl\\|\{b_n\}_{n=1}^\infty\bigr\\|_{\ell^q}\leq M\). (HW 17.2) State and prove a Riesz representation theorem for the bounded linear functionals on the sequence spaces \(\ell^p\), \(1\leq p\lt\infty\). (HW 17.3) Let \(c_0\) be the collection of all bounded sequences \(\{a_n\}_{n=1}^\infty\) such that \(\lim_{n\to\infty} a_n=0\). Equip \(c_0\) with the \(\ell^\infty\)-norm, that is, \(\\|\{a_n\}_{n=1}^\infty\\|_{c_0} = \\|\{a_n\}_{n=1}^\infty\\|_{\ell^\infty} = \sup_{n\in\mathbb{N}} \|a_n\|\). Determine the dual space to \(c_0\). (Make sure you use the condition \(\lim_{n\to\infty} a_n=0\); the dual space to \(\ell^\infty\) is not the same as the dual space to \(c_0\).) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 18	Tuesday, Feb. 11, 2025, 11:59 p.m.	(HW 18.1) Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive numbers with \(a_n\to\infty\). Let \(X\) be a normed linear space. Suppose that \(X\) contains an unbounded weakly convergent sequence; that is, suppose that there is an \(x\in X\) and a sequence \(\{x_n\}_{n=1}^\infty\) such that \(x_n\rightharpoonup x\) in \(X\) and \(\{\\|x_n\\|\}_{n=1}^\infty\) is unbounded. Show that there exists a \(y\in X\) and a sequence \(\{y_n\}_{n=1}^\infty\) in \(X\) with \(\\|y_n\\|=a_n\) and with \(y_n\rightharpoonup y\). Hint: The easiest choice of \(y\) is \(y=0\). (HW 18.2) Let \(\{a_k\}_{k=1}^\infty\) be a sequence in a finite dimensional real vector space \((X,\\|\,\\|).\) Suppose that \(\{a_k\}_{k=1}^\infty\) converges weakly in \(X\) to \(a\) for some \(a\in X.\) Show that \(\{a_k\}_{k=1}^\infty\) converges strongly in \(X\) to \(a.\) Do not assume that \(X\) is an inner product space; that is, do not assume that the norm on \(X\) can be written as the usual Pythagorean norm. Hint: Let \(\{e_i\}_{i=1}^n\) be a basis for \(X\) and define \(\\|\sum_{i=1}^n a_ie_i\\|_2=\left(\sum_{i=1}^n \|a_i\|^2\right)^{1/2}\). It is clear (you may use this fact) that \(\\|\,\\|_2\) is another norm on \(X\). Start by showing that this norm is equivalent to the original norm, that is, that there exist real numbers \(M\), \(m\) with \(0\lt m\leq M\) such that \(m\\|x\\|_2\leq \\|x\\|\leq M\\|x\\|_2\). Then show that a sequence converges (weakly or strongly) in \((X,\\|\,\\|)\) if and only if it converges (weakly or strongly) in \((X,\\|\,\\|_2)\). (HW 18.3) Let \(1\lt p\lt \infty\). For each \(n\in\mathbb{N}\), let \(\zeta_n=\{\zeta_{n,k}\}_{k=1}^\infty\) be an element of \(\ell^p\) and assume that \(\sup_{n\in\mathbb{N}} \\|\zeta_n\\|_{\ell^p}\lt\infty.\) Let \(\xi=\{\xi_k\}_{k=1}^\infty\) be another element of \(\ell^p\). Show that \(\zeta_n\rightharpoonup \xi\) weakly in \(\ell^p\) if and only if \(\lim_{n\to\infty} \zeta_{n,k}=\xi_k\) for all \(k\in\mathbb{N}\). Give an example of a sequence of elements of \(\ell^p\) that converges weakly but not strongly. A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 19	Tuesday, Feb. 18, 2025, 11:59 p.m.	(HW 19.1) Let \(E\subseteq\mathbb{R}\) be measurable and let \(2\lt p\lt \infty\). Show that, if \(a\), \(b\in\mathbb{R}\), then \[\biggl\|\frac{a+b}{2}\biggr\|^p+\biggl\|\frac{a-b}{2}\biggr\|^p \leq \biggl(\biggl\|\frac{a+b}{2}\biggr\|^2+\biggl\|\frac{a-b}{2}\biggr\|^2\biggr)^{p/2} \leq \frac{1}{2}\|a\|^p+\frac{1}{2}\|b\|^p.\] Prove Clarkson's inequality: if \(f\), \(g\in L^p(E)\), then \[\left\\| \frac{f + g}{2} \right\\|_{L^p(E)}^p + \left\\| \frac{f - g}{2} \right\\|_{L^p(E)}^p \le \frac{1}{2} \left( \\| f \\|_{L^p(E)}^p + \\| g \\|_{L^p(E)}^p \right).\] Show that \(L^p(E)\) is uniformly convex, that is, show that if \(\varepsilon\gt 0\) then there is a \(\delta\gt 0\) such that, if \(\\|f\\|_p=1=\\|g\\|_p\) and \(\\|f-g\\|_p\geq \varepsilon\), then \(\left\\|\frac{f+g}{2}\right\\|_p\leq 1-\delta\). (HW 19.2) Let \(E\subseteq\mathbb{R}\) be measurable and let \(2\lt p\lt \infty\). Show that if \(\{f_n\}_{n=1}^\infty\) is a sequence of functions in \(L^p(E)\) that satisfies \(\\|f_n\\|_p=1\) for all \(n\in\mathbb{N}\) and \(f_n\rightharpoonup f\) weakly in \(L^p(E)\) for some \(f\in L^p(E)\) with \(\\|f\\|_p=1\), then \(\{f_n\}_{n=1}^\infty\) is a Cauchy sequence. Prove the Radon-Riesz theorem for \(p\gt 2\). (HW 19.3) Without using any of the results of Section 8.4 (in particular, without using Lemma 8.16, Theorem 8.17, or Corollary 8.18), show that if \(E\subseteq\mathbb{R}\) is measurable, \(1\lt p\lt\infty\), and \(T:L^p(E)\to\mathbb{R}\) is a continuous linear functional, then there is a \(f_0\in \{f\in L^p(E):\\|f\\|_p\leq 1\}\) such that \(T(f_0)\leq T(f)\) for all \(f\in \{f\in L^p(E):\\|f\\|_p\leq 1\}\). A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 20	Tuesday, Feb. 25, 2025, 11:59 p.m.	(HW 20.1) Find a sequence \(\{g_n\}_{n=1}^\infty\) in \(L^2[0,1]\) that has no Cauchy subsequence. Then use this subsequence and Section 7.4 to find a sequence of continuous functions on \([0,1]\) such that no subsequence converges in \(L^2[0,1]\) to a function in \(L^2[0,1]\). (HW 20.2) Let \((X,\mathcal{M})\) be a measurable space. Suppose that \(\mu\) and \(\nu\) are two measures on \(\mathcal{M}\) and that \(\mu(S)\geq \nu(S)\) for all \(S\in\mathcal{M}\). (We abbreviate this condition as \(\mu\geq \nu\).) Show that there exists a measure \(\lambda\) on \(\mathcal{M}\) such that \(\mu(S)=\nu(S)+\lambda(S)\) for all \(S\in\mathcal{M}\) (we abbreviate this condition as \(\mu=\nu+\lambda\)). If \(\nu\) is \(\sigma\)-finite, show that there is exactly one such \(\lambda\). If \(\nu\) is not \(\sigma\)-finite, show that there is a smallest such \(\lambda\), that is, a \(\lambda\) such that \(\mu=\nu+\lambda\) and such that \(\lambda\leq \theta\) for every measure \(\theta\) on \(\mathcal{M}\) that satisfies \(\mu=\nu+\theta\). Give an example of a measurable space \((X,\mathcal{M})\) and measures \(\mu\), \(\nu\), \(\lambda\), and \(\theta\) on \(\mathcal{M}\) such that \(\lambda(S)\neq \theta(S)\) for at least one \(S\in\mathcal{M}\) but such that \(\mu(S)=\nu(S)+\lambda(S)=\nu(S)+\theta(S)\) for all \(S\in\mathcal{M}\). (HW 20.3) Let \(X\) and \(Y\) be sets with \(Y\subset X\) and with \(\emptyset\neq Y\neq X\). Let \(0\lt\alpha\lt\infty\), let \(S=\{Y\}\), and let \(\mu:S\to[0,\infty]\) be given by \(\mu(Y)=\alpha\). Determine the induced outer measure \(\mu^\). Determine the \(\sigma\)-algebra \(\mathcal{M}\) of sets that are measurable with respect to \(\mu^\). A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 21	Tuesday, Mar. 4, 2025, 11:59 p.m.	(HW 21.1) (HW 21.2) (HW 21.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 22	Tuesday, Mar. 11, 2025, 11:59 p.m.	(HW 22.1) (HW 22.2) (HW 22.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 23	Tuesday, Mar. 18, 2025, 11:59 p.m.	(HW 23.1) (HW 23.2) (HW 23.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
	Mar. 24–28	Spring break (no class)
HW 24	Tuesday, Apr. 1, 2025, 11:59 p.m.	(HW 24.1) (HW 24.2) (HW 24.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 25	Tuesday, Apr. 8, 2025, 11:59 p.m.	(HW 25.1) (HW 25.2) (HW 25.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 26	Tuesday, Apr. 15, 2025, 11:59 p.m.	(HW 26.1) (HW 26.2) (HW 26.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 29	Thursday, May. 8, 2025, 11:59 p.m.	(HW 29.1) (HW 29.2) (HW 29.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
	Friday, Apr. 18	Last day to withdraw from a course.
HW 27	Tuesday, Apr. 22, 2025, 11:59 p.m.	(HW 27.1) (HW 27.2) (HW 27.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 28	Tuesday, Apr. 29, 2025, 11:59 p.m.	(HW 28.1) (HW 28.2) (HW 28.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
	Date TBA	Complete the online course evaluation on or before the due date. If at least 75% of the class completes the course evaluation before the deadline, or if the registrar cancels course evaluations due to low enrollment, I will drop your 2 lowest homework scores; otherwise, I will drop your 1 lowest homework score.
	Wednesday, Apr. 30	Penultimate day of class
HW 29	Thursday, May 8, 2025, 11:59 p.m.	(HW 29.1) (HW 29.2) (HW 29.3) A LaTeX template for this homework may be found here. General advice on LaTeX and Overleaf may be found here.
HW 30	Wednesday, May 7, 12:45 a.m.–2:45 p.m.	Attend class at the date and time reserved for our final exam (as indicated on UAConnect or the registrar’s website) and make a good faith effort to present at least one problem. This is the only day that you are required to attend class and also the only day you are required to present your problem. Any presentation points earned may in addition be applied to the problems on HW 29.

Course policies

Instructor: Ariel Barton

GA tutor: Emily Freeman

Contact information:
Email: aeb019@uark.edu
Office: SCEN 348
Email will be reviewed within one business day.

Class time and location: Monday, Wednesday, and Friday, 11:50 a.m.–12:40 p.m., January 13–Apr. 30, 2025, Kimpel Hall 116.

Office hours: No office hours will be held January 22--24 or over spring break. In all other weeks, office hours will be held Office hours will be held at

Tuesdays, 2:00, SCEN 419 (Emily Freeman)
Wednesdays, 3:05, SCEN 348 (Ariel Barton)
Thursdays, 2:00, Zoom (Ariel Barton; I will be in SCEN 348 if I am on campus)
Fridays, 3:05, SCEN 348 (Ariel Barton)
Fridays, by appointment (Emily Freeman, egf005@uark.edu)

These office hours are on a drop in basis; at these times, you do not need an appointment.

If the selected times do not work for you, or if you wish to speak to me privately, please email me and we will schedule an appointment at another time.

Course Description: Measure and integration on abstract measure spaces, signed measures, Hahn decomposition, Radon-Nikdoym theorem, Lebesque decomposition, measures on algebras and their extensions, product measures, and Fubini's theorem.

Prerequisites: MATH 55003 or departmental consent.

Course Objectives: During this course, students should:

Learn the content and applications of the major theorems of real analysis.
Further develop their skills at formulating and writing proofs.
Develop skills at independent study.

Textbook: Real Analysis, fifth edition, by Halsey L. Royden and Patrick M. Fitzpatrick, ISBN 9780137906529/9780136853541. Students may also wish to consult Measure, Integration & Real Analysis by Sheldon Axler.

Inquiry-based learning: Students will be supplied with a list of lecture problems (PDF or HTML). This list of problems constitutes the notes for this course. Written homework assignments will be posted to this web page or Blackboard.

Students may choose to either present solutions to lecture problems during class, or to turn in written homework assignments.

Specifically, a student who presents a complete solution at the board during class will be granted 2–24 presentation points (depending on the problem and on originality of their solution). Students may view their remaining presentation points on Blackboard.

If a student does not turn in a written assignment or turns in an assignment with fewer than 3 problems worked, and has remaining presentation points to spend, then 12 presentation points will be deducted per unworked problem and the assignment score will be recorded as though the student had received full credit on the missing problem or problems.

Homework: Homework assignments will be posted to this web page and should be turned in through Blackboard. I expect to have 14 regular assignments throughout the semester. I expect to drop your 2 lowest homework scores; your course grade will be determined by your 12 highest homework scores. You may work together on the homework assignments, but each student must write up their work in their own words and submit their own work for grading. Graded homework will be returned via email.

Homework rewrites: You can expect to have three problems on each of your weekly homework assignments. On each homework assignment, you are permitted to rewrite one problem (not two problems or all three problems) whose grade you are unhappy with. The rewrite’s due date will be included in the email you receive returning your homework to you and will generally be at least five days later. Homework may be uploaded to the same location in Blackboard as the original assignment.

Homework formatting: Homework submission is generally required to follow the following rules. If you feel that they place an undue burden on you (for example, if you do not own a computer), talk to me and we will arrange an exception or accommodation.

It should be in PDF format (a compiled PDF, not TeX source code).

My annotation software (the software I use for grading) can deal with PDF files better than almost any other format. Also, it is generally straightforward to convert other formats to PDF and my grading is much smoother if all homework files are in the same format.
It must be easy to read, and the file size must not be too big. This generally means that you must type your homework using LaTeX. Here is a quick guide to LaTeX for those unfamiliar with the program. The following template produces output that I find easy to read and grade.
It should uploaded to the correct location in Blackboard.

This collects all students’ homework in a standardized location (meaning I am less likely to lose your homework) and also gives your homework standardized file names (which makes it much easier to return). If you are unable to upload your homework to Blackboard, instead email it to me. The file name must be HW-N-username.pdf (or Rewrite-N-username.pdf), where N is the homework number and username is your UA username (that is, your university email address without the @uark.edu).

Exams and course grades: There will be no exams in this course unless the majority of students request that exams be held. Your course grade will be based entirely on your homework scores (including presentation points as indicated above). This means that we will hold an (otherwise as usual) two-hour class at the date and time reserved for our final exam as indicated on the registrar’s website.

Electronic devices: Cell phones, tablets, laptops, and other electronic devices may be used in class. The expectation is that these devices will be used for taking notes, routine calculations (i.e., calculator apps), accessing course materials, and other course-related uses only. Please do not text or play games in class!

Inclement weather policy: Class will meet unless the University is closed. On-campus students are expected to be present. Off-campus students should make their own decisions in the best interest of personal safety. Off-campus students will not be penalized for being absent on those days the Fayetteville Public Schools are closed due to weather, but are encouraged to attend by Zoom if possible.

If the University is closed due to inclement weather, office hours will be held over Zoom unless the instructor is without electricity or internet access. Class will either be also held over Zoom or cancelled; you will be notified of which by email as soon as possible.

If attendance is severely affected by weather, deadlines and exam dates may be adjusted. Please do not call the Department of Mathematical Sciences with weather-related inquiries. You may contact me for information.

Recording of Class Lectures: The lecture will be recorded on the room’s camera and recordings will be posted to Blackboard for student reference. These recordings are provided “as is” as a hopefully-useful supplement, but their availability and quality is not guaranteed and no make-up recordings or materials will be posted in the event of technical issues (such as the room's camera recording the wrong wall).

Students who need or particularly desire high-quality reliable recordings are allowed to record lectures with their own devices for their own use.

In addition, if the University is open but the weather is severe, a Zoom meeting will be opened during class so that students may participate remotely in real time.

By attending this class, the student understands the course may be recorded and consents to being recorded for official university educational purposes. Be aware that incidental recording may also occur before and after official class times. Recordings may include personally-identifiable comments submitted to the chat stream during class.

Department and University policies

Etiquette: Students and instructors each have an important role in maintaining a classroom environment optimal for learning, and are expected to treat each other with respect during class, using thoughtful dialogue, and keeping disruptive behaviors to a minimum. Both students and faculty perceive abusive language directed towards others as the most disruptive behavior. Other behaviors that can be disruptive are chatting and whispering during class, the use of smartphones or laptops for texting or in other ways unrelated to the course, preparing to leave before class is over, and consistently arriving late to class. Inappropriate behavior in the classroom may result in a request to immediately leave the classroom and/or a referral to the Office of Academic Integrity and Student Conduct.

Academic honesty: Academic dishonesty will not be tolerated. “As a core part of its mission, the University of Arkansas provides students with the opportunity to further their educational goals through programs of study and research in an environment that promotes freedom of inquiry and academic responsibility. Accomplishing this mission is only possible when intellectual honesty and individual integrity prevail.” Each University of Arkansas student is required to be familiar with and abide by the University's Academic Integrity Policy, which may be found https://honesty.uark.edu/policy/index.php. Students with questions about how these policies apply to a particular course or assignment should immediately contact their instructor.

Inclement Weather Policy: Class will meet unless the University is closed. On-campus students are expected to be present. Off-campus students should make their own decisions in the best interest of personal safety. Off-campus students will not be penalized for being absent on those days the Fayetteville Public Schools are closed due to weather. If attendance is severely affected by weather, deadlines and exam dates may be adjusted. Please do not call the Department of Mathematical Sciences with weather-related inquiries. You may contact me for information.

Accommodations: Under University policy and federal and state law, students with documented disabilities are entitled to reasonable accommodations to ensure the student has an equal opportunity to perform in class. If any member of the class has such a disability and needs special academic accommodations, please report to the Center for Educational Access (CEA). Reasonable accommodations may be arranged after the CEA has verified your disability. Students who are registered with the Center for Educational Access must meet with the instructor by the end of the first week of class, or within one week of registering with CEA to discuss their accommodations. This must be done before you utilize your accommodations. Do not hesitate to contact your instructor if any assistance is needed in this process.

Communication: The best way to communicate with me (beyond coming to my office hours) is by email. You can expect a reply within 24 hours, excluding weekends and holidays. Students are expected to monitor their uark email address regularly and consult the course Blackboard site for important announcements.

Intellectual Property: Notes, review material, exams, quizzes, videos or other learning material used in this class are the intellectual property of the instructor. Selling or freely sharing this content in electronic or written form is a violation of intellectual property rights and also constitutes a violation of the University’s academic integrity policy. Your continued enrollment in this class signifies your understanding of and your intent to abide by this policy. There are severe consequences for sharing class content online.

Emergency Procedures: Many types of emergencies can occur on campus; instructions for specific emergencies such as severe weather, active shooter, or fire can be found at https://safety.uark.edu/emergency-preparedness/.

Health and wellness: https://catalog.uark.edu/generalinfo/studentaffairs/#healthcentertext

Grade Disputes: The instructor is committed to keeping students informed of their standing in the class. Scores on all graded items will be posted in a timely manner. Students are expected to bring any possible errors to the attention of the instructor within one week of posting. This maintains an accurate grade record throughout the semester. All scores posted before Dead Day will be deemed accurate unless a possible error is brought to the attention of the instructor before the scheduled final exam.

Class Cancellation Policies & Procedures:

Class cancellation by the University due to inclement weather-related delay/closure: In the event of an inclement weather delay, early dismissal, or closure, the instructor may conduct class over Zoom at the scheduled class time while the campus is closed for inclement weather but will not require a student to attend synchronously. Any class taught remotely while the campus is closed for inclement weather will be recorded and made available to all students in the class in the approved campus Learning Management System. Your instructor will provide students the opportunity to make up work due while the University is closed for inclement weather without penalty. If remote instruction is not used, the instructor will cancel class meetings altogether. Instead, alternative learning materials and assignments will be supplied, to make up the missing class days.
Faculty absence due to medical or other emergency: Every effort will be made to arrange for a substitute if your instructor must take an emergency leave. If a substitute cannot be arranged, then your instructor may choose to teach classes remotely when they are able to do so, subject to appropriate university policies on remote instruction. If your instructor cancels class in case of an emergency (allowed per university policy https://provost.uark.edu/policies/185810.php), then your instructor will make up missed class time using recorded lectures, assignments, readings, instructional materials, or other alternative forms of instruction. Your instructor will send announcements promptly via email and posted to Blackboard. Each student will also get the opportunity to make up work that is due if class is impacted by an emergency.

Unauthorized Use of Class Recordings: These recordings may be used by students only for the purposes of the class. Students may not download, store, copy, alter, post, share, or distribute in any manner all or any portion of the class recording, (e.g. a 5- second clip of a class recording sent as a private message to one person is a violation of this provision). This provision may protect the following interests (as well as other interests not listed): faculty and university copyright; FERPA rights; and other privacy interests protected under state and/or federal law. Unauthorized recording, or transmission of a recording, of all or any portion of a class is prohibited unless the recording is necessary for educational accommodation as expressly authorized and documented through the Center for Educational Access with proper advance notice to the instructor. Unauthorized recordings may violate federal law, state law, and university policies. Student-made recordings are subject to the same restrictions as instructor-made recordings. Failure to comply with this provision will result in a referral to the Office of Student Standards and Conduct for potential charges under the Code of Student Life. In situations where the recordings are used to gain an academic advantage, it may also be considered a violation of the University of Arkansas’ academic integrity policy.

Incompletes: Only given in extreme circumstances, and only when the student has satisfactorily completed all but a small portion of the work in the course. Students must make prior arrangements with the professor well before the end of the semester.

Math 55103Theory of Functions of a Real Variable IISpring 2025

Homework assignments and important dates

Course policies

Department and University policies

Math 55103
Theory of Functions of a Real Variable II
Spring 2025