## MATH4331 real analysis课程简介

The study of metric spaces and real-valued functions is an important branch of mathematical analysis. In this field, mathematicians explore the properties of mathematical structures known as metric spaces, which consist of a set of points along with a distance function that assigns a distance between any two points in the set. These structures are used to define and study various types of functions, such as real-valued functions, which map points in a metric space to real numbers.

One of the central concepts in this field is the convergence of a sequence of points in a metric space. A sequence is said to converge to a limit if its distance from the limit point becomes arbitrarily small as the sequence progresses. This concept is used to define important properties of functions such as continuity and uniform continuity. A function is said to be continuous if the limit of the function at a given point is equal to the function’s value at that point. A function is uniformly continuous if the difference between the function’s values at two points becomes arbitrarily small as the distance between those points becomes arbitrarily small.

## PREREQUISITES

Sequences and series of functions are also studied in this field. For example, a sequence of functions can be said to converge pointwise if the limit of the sequence at each point in the domain exists. Alternatively, the sequence can be said to converge uniformly if the difference between the function values becomes arbitrarily small uniformly across the domain.

Differentiation and Riemann-Stieltjes integration are two additional topics that are explored in this field. The derivative of a function measures the rate at which the function changes at a given point, while the Riemann-Stieltjes integral generalizes the Riemann integral to allow for integration with respect to a more general class of functions.

Overall, the study of metric spaces and real-valued functions is a rich and important field that has many applications in mathematics and other areas of science and engineering.

## MATH4331 real analysis HELP（EXAM HELP， ONLINE TUTOR）

(1) Define differentiability of a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$.

Differentiability is a key concept in calculus and analysis that allows us to measure how much a function changes over a small interval. A function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is said to be differentiable at a point $\mathbf{x}_0 \in \mathbb{R}^n$ if there exists a linear transformation $L:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that

\lim_{\mathbf{x} \to \mathbf{x}_0} \frac{\left\|f(\mathbf{x}) – f(\mathbf{x}_0) – L(\mathbf{x} – \mathbf{x}_0) \right\|}{\|\mathbf{x} – \mathbf{x}_0\|} = 0,xx0​lim​∥xx0​∥∥f(x)−f(x0​)−L(xx0​)∥​=0,

where $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^n$. The linear transformation $L$ is called the derivative of $f$ at $\mathbf{x}_0$, denoted by $Df(\mathbf{x}_0)$ or $f'(\mathbf{x}_0)$.

Intuitively, the derivative of a function at a point measures the instantaneous rate of change of the function at that point. This rate of change is represented by the linear transformation $L$, which is a matrix in the case of functions from $\mathbb{R}$ to $\mathbb{R}$, and a Jacobian matrix in the case of functions from $\mathbb{R}^n$ to $\mathbb{R}^n$. The derivative also provides important information about the behavior of the function near the point of differentiability, such as the local slope of the tangent plane to a surface or the direction of steepest ascent of a scalar-valued function.

(2) State the contraction mapping principle.

The contraction mapping principle, also known as the Banach fixed-point theorem, is a fundamental result in the theory of metric spaces. It states that any contraction mapping on a complete metric space has a unique fixed point, which is the point that the mapping maps to itself.

More formally, let $(X,d)$ be a complete metric space, and let $T:X \rightarrow X$ be a contraction mapping with a contraction factor $0 \leq k < 1$. That is, for any $x,y \in X$, we have

d(T(x), T(y)) \leq k d(x,y).d(T(x),T(y))≤kd(x,y).

Then, there exists a unique fixed point $x^* \in X$ such that $T(x^) = x^$.

Intuitively, the contraction mapping principle tells us that if a function $T$ “shrinks” the distance between its inputs by a factor of $k$, and $X$ is complete, then there must exist a unique point $x^*$ that is mapped to itself under $T$. This result has important applications in many fields, including optimization, numerical analysis, and economics.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.