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 $\mathfrak{T}$ (...
Posts by Tag
- open set 3
- closed set 3
- Approximation of the identity 3
- completeness 2
- uncountable 2
- random variable 2
- derived set 2
- closure 2
- compact 2
- continuity 2
- jump discontinuity 2
- differentiation 2
- Darboux Theorem 2
- Riemann Integral 2
- uniform convergence 2
- set 1
- De Morgan's law 1
- well-ordering principle 1
- mathematical induction 1
- supremum 1
- infimum 1
- Archimedean property 1
- dense 1
- root 1
- countable 1
- Cantor's diagonal process 1
- triangular inequality 1
- convergent sequence 1
- limit theorems 1
- squeeze theorem 1
- monotone sequence 1
- monotone convergence theorem 1
- nested interval property 1
- limit point 1
- isolated point 1
- Bolzano-Weierstrass Theorem 1
- limsup 1
- liminf 1
- probability space 1
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- density function 1
- mass function 1
- fundamental theorem 1
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- Cantor set 1
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- intermediate value theorem 1
- uniform continuity 1
- Lipschitz continuity 1
- discontinuity 1
- montone function 1
- derivative 1
- chain rule 1
- Mean Value Theorem 1
- Rolle's Theorem 1
- Intermediate Value Theorem 1
- Inverse Function Theorem 1
- L'Hôpital's rule 1
- Taylor's Theorem 1
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- Riemann Criterion 1
- Composition Theorem 1
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open set
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 $N_{\epsilon} (p) \subset E$. ...
closed set
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.
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 $N_{\epsilon} (p) \subset E$. ...
Approximation of the identity
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 $[-\pi, \pi]$ ...
completeness
Cauchy Sequence
Definition Let \(\{p_n\}_{n=1}^\infty\) be a sequence in $\mathbb{R}$. The sequence is a Cauchy sequence if $\forall \epsilon >0, \exists N\in \mathbb{N}$...
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
uncountable
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_{...
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 \...
random variable
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...
derived set
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 $N_{\epsilon} (p) \subset E$. ...
closure
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.
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 $N_{\epsilon} (p) \subset E$. ...
compact
Continuous functions-(2)
Lemma 1
Compact set
Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded
continuity
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...
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(...
jump discontinuity
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 ...
differentiation
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 ...
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 ...
Darboux Theorem
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 $a,b\in I$...
Riemann Integral
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...
uniform convergence
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...
set
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$
De Morgan's law
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$
well-ordering principle
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$
mathematical induction
Set and Function
Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$
supremum
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
infimum
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
Archimedean property
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
dense
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
root
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
countable
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 \...
Cantor's diagonal process
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 \...
triangular inequality
Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
convergent sequence
Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
limit theorems
Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
squeeze theorem
Sequences of real numbers
Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$
monotone sequence
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_...
monotone convergence theorem
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_...
nested interval property
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_...
limit point
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...
isolated point
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...
Bolzano-Weierstrass Theorem
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...
limsup
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
liminf
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
probability space
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.
distribution function
Distribution Function
Distribution Function Let $X$ be a random variable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$. The distribution function (cumulative distribution functi...
density function
Distribution Function
Distribution Function Let $X$ be a random variable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$. The distribution function (cumulative distribution functi...
mass function
Distribution Function
Distribution Function Let $X$ be a random variable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$. The distribution function (cumulative distribution functi...
fundamental theorem
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...
isomorphism
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...
linear transformation
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...
Cauchy Sequence
Cauchy Sequence
Definition Let \(\{p_n\}_{n=1}^\infty\) be a sequence in $\mathbb{R}$. The sequence is a Cauchy sequence if $\forall \epsilon >0, \exists N\in \mathbb{N}$...
connected
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...
Bolzano-Weierstrass property
Compact set
Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded
Heine-Borel Theorem
Compact set
Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded
Heine-Borel-Bolzano-Weierstrass Theorem
Compact set
Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded
Cantor set
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_{...
measure zero
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_{...
limit of function
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...
epsilon-delta
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...
bounded
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...
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 ...
quotient 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 ...
normal subgroup
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 ...
ideal
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 ...
quotient 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 ...
well-defined binary operation
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 ...
topological characterization
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(...
min-max value theorem
Continuous functions-(2)
Lemma 1
intermediate value theorem
Continuous functions-(2)
Lemma 1
uniform continuity
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 ...
Lipschitz continuity
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 ...
discontinuity
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...
montone function
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...
derivative
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 ...
chain rule
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 ...
Mean Value Theorem
Mean Value Theorem
Definition 5.2.1 Let $E\subset \mathbb{R}$ be a set and let $f:E\to\mathbb{R}$.
Rolle's Theorem
Mean Value Theorem
Definition 5.2.1 Let $E\subset \mathbb{R}$ be a set and let $f:E\to\mathbb{R}$.
Intermediate Value Theorem
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 $a,b\in I$...
Inverse Function Theorem
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 $a,b\in I$...
L'Hôpital's rule
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...
Taylor's Theorem
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...
Integration
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...
upper sum
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...
lower sum
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...
Riemann Criterion
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...
Composition Theorem
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 $\varphi:[c,d]\to\mathbb{R}$ ...
comparison test
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_...
root test
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_...
ratio test
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_...
Abel Partial Summation Formula
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 $1\leq p \leq q$,
Dirichlet Test
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 $1\leq p \leq q$,
Absolute convergence
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 $1\leq p \leq q$,
Rearrangement of series
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 $1\leq p \leq q$,
Sequence of functions
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...
Series of functions
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...
Pointwise convergence
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...
Uniform convergence
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...
Cauchy criterion
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...
Weierstrass M-Test
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...
Dini Theorem
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...
integration
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 ...
boundedness
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 $\{f_n(x)\}$...
equicontinuous
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 $\{f_n(x)\}$...
Ascoli-Azrela Theorem
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...
Stone-Weirstrass Theorem
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...
Analytic function
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...
basis
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 $\mathfrak{T}$ (...
product topology
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...
subspace
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...
interior
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.
limit points
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.
dimension theorem
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...
rank
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...
rank theorem
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...
Lebesgue integral
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...
Lebesgue monotone convergence theorem
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...
pseudo-inverse - linear transformation
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...
Fourier Sereis
Introduction to Fourier Series
Basic Knowledge We focus on a class of functions
Dirichlet kernel
Introduction to Fourier Series
Basic Knowledge We focus on a class of functions
Poisson kernel
Introduction to Fourier Series
Basic Knowledge We focus on a class of functions
Convolution
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 $[-\pi, \pi]$ ...
Good Kernels
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 $[-\pi, \pi]$ ...
Fejer's Theorem
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...
Fejer Kernel
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...
Dirichlet Problem
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...
Polar Coordinates
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...
Hilbert Space
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
Best Approximation Lemma
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
Infinite dimensional vector space
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
Parseval's identity
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 ...
Polarized Parseval's identity
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 ...
Riemann-Lebesgue Lemma
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 ...
Localization principle of Riemann
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 ...
Sawtooth Function
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 $f(0)=0$, ...
Divergent Fourier series of continuous function
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 $f(0)=0$, ...
Curve
Fourier series and the isoperimetric inequality
Definition 1.1 A $\mathscr{C}^1$ mapping
Isoperimetric Inequality
Fourier series and the isoperimetric inequality
Definition 1.1 A $\mathscr{C}^1$ mapping
Number theory
Weyl’s Equidistribution Theorem
Definition 1.1 Let $x\in\mathbb{R}$ be a real number. Then
Weyl's Equidistribution theorem
Weyl’s Equidistribution Theorem
Definition 1.1 Let $x\in\mathbb{R}$ be a real number. Then
continuous
Continuous but nowhere differentiable function
Theorem If $0<\alpha <1$, then the function
non differentiable
Continuous but nowhere differentiable function
Theorem If $0<\alpha <1$, then the function