Ascoli-Azrela and Stone-Weirstrass Theorem
Theorem (Ascoli-Azrela) Let $\{f_n\}$ be a sequence of functions defined on a set $K$. If $K$ is compact, $\{f_n\}$ is pointwise bounded on $K$, $\{f_n\}$ is...
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- Analysis 30
- Fourier Analysis 10
- Statistics 3
- Linear Algebra 3
- Topology 3
- Group 1
- Ring 1
- Measure Theory 1
Analysis
Equicontinuous Families of Functions
Definition 7.19 Let $\{f_n\}$ be a sequence of functions defined on a set $E$. We say that $\{f_n\}$ is pointwise bounded on $E$ if the sequence $\{f_n(x)\}$...
Uniform Convergence, Integration, and Differentiation
Theorem 8.4.1 Suppose $f_n\in \mathscr{R}[a,b]$ for all $n\in\mathbb{N}$ ans suppose that the sequence $\{f_n\}$ converges uniformly to $f$ on $[a,b]$. Then ...
Uniform Convergence and Continuity
Theorem 8.3.1 Suppose $\{f_n\}$ is a sequence of real-valued functions that converges uniformly to a function $f$ on a subset $E$ of $\mathbb{R}$. Let $p$ be...
Pointwise Convergence and Uniform Convergence
Definition 8.1.1 (Pointwise Convergence) A sequence of real valued functions $\{f_n\}$ defined on a set $E (\subset \mathbb{R})$ converges pointwise on $E$ i...
Dirichlet Test and Absolute Convergence
Theorem 7.2.1 Let $\{a_k\}$ and $\{b_k\}$ be a sequence of real numbers. Set $A_0:=0, A_n:=\sum_{k=1}^n a_k$ for $n\geq 1$. Then if $1\leq p \leq q$,
Convergence Test
Definition Let $\{a_n\}$ be a sequence. Define a new sequence of $s_n := \sum_{k=1}^na_k$ for $n\geq 1$. We call the $s_n$ is the $n$-th partial sum of $\{a_...
Properties of the Riemann Integral
Theorem 6.1.9 Let $f:[a,b]\to\mathbb{R}$ be a bounded real valued Riemann integrable function with $Range f \subset [c,d]$. Let $\varphi:[c,d]\to\mathbb{R}$ ...
Riemann Integral
Definition (upper sum, lower sum) Let $[a,b]$ with $a<b$ be a closed and bounded interval in $\mathbb{R}$. By a partition of $\mathscr{P}$ of $[a,b]$ we m...
Continuity of the Derivative
Darboux Theorem $f:[a,b]\to\mathbb{R}$ is differentiable on $[a,b]$. Assume that $f^\prime(a)<f^\prime(b)$. Then $\forall\lambda \in (f^\prime(a), f^\prim...
Intermediate Value Theorem for derivatives
Theorem 4.2.13 (Intermediate Value Theorem for derivative) Let $f:I\to\mathbb{R}$ be differentiable function on the interval $I=[a,b]$. Then given $a,b\in I$...
Mean Value Theorem
Definition 5.2.1 Let $E\subset \mathbb{R}$ be a set and let $f:E\to\mathbb{R}$.
Differentiation
Definition 5.1.1 Let $I\subset \mathbb{R}$ be an interval and let $f:I\rightarrow\mathbb{R}$ be a function. Fix a $p\in I$. The derivative of $f$ at $p$ is ...
Monotone Functions and Discontinuities
Definition 4.4.1 $E\subset \mathbb{R}, f:E\to\mathbb{R}, p$ is a limit point of $E\cap (p,\infty)$. $f$ has a right limit at $p$ if there exists a $L\in\math...
Uniform Continuity
Definition 4.3.1 Let $E$ be a subset of $\mathbb{R}$ and let $f:E\to\mathbb{R}$ be a function. The function $f$ is uniformly continuous on $E$ if $\forall ...
Continuous functions-(2)
Lemma 1
Continuous functions-(1)
Definition 4.2.1 $f:E\to \mathbb{R}$ is continuous at $p\in E$, if $\forall \epsilon >0, \exists \delta >0$ such that $|x-p|<\delta \Rightarrow |f(...
Limit of function
4.1.1 Definition $E \subset \mathbb{R}, f: E \rightarrow \mathbb{R}$: a function, $p \in E^\prime$. $f$ has a limit at p if there exists $L\in \mathbb{R}$ su...
Cantor Set
Definition of Cantor Set For each $P_i$ is non empty compact set and $P_0 \supset P_1 \supset P_2 \cdots$. Define $P$ as follows: \(\begin{align} P := \cap_{...
Compact set
Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded
Open and closed set-(2)
Theorem 3.1.13 $U \subset \mathbb{R}$ is open $\Rightarrow$ \(\exists \{I_n\}\): a finite or countable family of pairwise disjoint union of open intervals su...
Open and closed set
Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...
Cauchy Sequence
Definition Let \(\{p_n\}_{n=1}^\infty\) be a sequence in $\mathbb{R}$. The sequence is a Cauchy sequence if $\forall \epsilon >0, \exists N\in \mathbb{N}$...
Limit Superior and Inferior of a Sequence
Definition 2.5.1 Let $\{s_n\}$ be a sequence in $\mathbb{R}$. The limit superior of $\{s_n\}$, denoted as $\limsup_{n\to\infty}s_n$, is defined as
Subsequences and Bolzano-Weierstrass Theorem
Definition 2.4.1 Let \(\{p_n\}\) be a sequence and let \(\{n_k\}\) be strictly increasing sequence, i.e. $n_1 < n_2 <n_3 < \cdots$. We call \(\{p_{n...
Monotone Sequences
Definition 2.3.1 A sequence \(\{a_n\}\) is said to be (a) monotone increasing if $a_n \leq a_{n+1}$ for all $n\in\mathbb{N}$ (b) monotone decreasing if $a_...
Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
Countable and Uncountable sets
Definition 1.7.2 For each positive integer $n\in \mathbb{N}$, let \(\mathbb{N}_n:=\{1,2,\ldots,n \}\). If $A$ is a set, we say (a) $A$ is finite if $A\sim \...
Least Upper Bound Property
Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$
Fourier Analysis
Continuous but nowhere differentiable function
Theorem If $0<\alpha <1$, then the function
Weyl’s Equidistribution Theorem
Definition 1.1 Let $x\in\mathbb{R}$ be a real number. Then
Fourier series and the isoperimetric inequality
Definition 1.1 A $\mathscr{C}^1$ mapping
Fourier series need not converge at points of continuity
Exercise 2.8 Verify that $\frac{1}{2i}\sum_{n\neq 0} \frac{e^{inx}}{n}$ is the Fourier series of the $2\pi$-periodic sawtooth function, defined by $f(0)=0$, ...
L2 Recovery of Integrable Functions and Consequences
Corollary 1.2 (Parseval’s Identity) Let $f$ be integrable function, and $a_n= \hat{f}(n)$. Then $\lim_{N\to\infty}\sum_{n=-N}^N\lvert a_n\rvert^2$ converges ...
Fourier Series in the Language of Infinite Dimensional Vector Space
Preliminaries Let $\{a_n: n \in\mathbb{Z}\}$ denote a sequence of complex numbers. We define $\ell_2$ norm of $\{a_n\}$ by
Dirichlet Problem on the Unit Disc
Preliminaries Suppose one has an infinite plate $(\mathbb{R}^2)$ with an initial heat distribution. Let $u(x,y)$ denote the temperature of the place at posit...
Cesaro and Abel Summability
Definition 1.1 Given a sequence $\{c_n\}$, let $s_n:=\sum_{k=0}^nc_k$ be the sequence of partial sums. We define $N$-th Cesaro mean $\sigma_N$ of the sequenc...
Convolution and Good Kernels
Definition 1.1 Let $f,g:\mathbb{R}\to\mathbb{C}$ be $2\pi$-periodic functions. The convolution $f*g$ of $f$ and $g$ is the function defined by $[-\pi, \pi]$ ...
Introduction to Fourier Series
Basic Knowledge We focus on a class of functions
Statistics
Distribution Function
Distribution Function Let $X$ be a random variable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$. The distribution function (cumulative distribution functi...
Random Variable
Measurable function Let $(E, \mathcal{E})$ and $(F, \mathcal{F})$ be measurable spaces where $E,F,$ are sets and $\mathcal{E}$ and $\mathcal{F}$ are $\sigma...
Probability Space
The sample space $\Omega$ is the set of all possible outcomes of an experiment. Points in $\omega \in \Omega$ are outcomes or realizations.
Linear Algebra
Moore-Penrose Pseudoinverse
Definition Let $V$ and $W$ be finite dimensional inner product spaces over the same field $F$ and let $T:V\to W$ be a linear transformation. Let $L:\ker T^\p...
Dimension Theorem and Rank Theorem
Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl...
Fundamental Theorem of Linear Algebra
Notation Let \(\mathfrak{B}=\{\mathbf{v}_1, \ldots, \mathbf{v}_n \} \text{ is a basis for } V.\) Define a function \([\cdot]_{\mathfrak{B}}:V \rightarr...
Topology
Closed Sets and Limit Points
Definition (Closed) A subset $A$ of a topological space $X$ is said to be closed if the set $X-A$ is open.
Product Topology and Subspace Topology
Definition (Product Topology on $X \times Y$) Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be topological spaces. The product space topology on $X\times...
Topology and Basis
Definition (Topology) A topology on a set $X$ is a collection $\mathfrak{T}$ having the following properties: (1) $\emptyset$ and $X$ are in $\mathfrak{T}$ (...
Group
Group and Ring
Definition 1 $\cdot: G\times G \to G$ is binary operation where we write $x\cdot y =xy$ for $x,y\in G$. $(G,\cdot)$ is group if it satisfies the followings ...
Ring
Group and Ring
Definition 1 $\cdot: G\times G \to G$ is binary operation where we write $x\cdot y =xy$ for $x,y\in G$. $(G,\cdot)$ is group if it satisfies the followings ...
Measure Theory
Lebesgue Integration
Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...