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
- distribution function 1
- density function 1
- mass function 1
- fundamental theorem 1
- isomorphism 1
- linear transformation 1
- Cauchy Sequence 1
- connected 1
- Bolzano-Weierstrass property 1
- Heine-Borel Theorem 1
- Heine-Borel-Bolzano-Weierstrass Theorem 1
- Cantor set 1
- measure zero 1
- limit of function 1
- epsilon-delta 1
- bounded 1
- group 1
- quotient group 1
- normal subgroup 1
- ring 1
- ideal 1
- quotient ring 1
- well-defined binary operation 1
- topological characterization 1
- min-max value theorem 1
- 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
- Integration 1
- upper sum 1
- lower sum 1
- Riemann Criterion 1
- Composition Theorem 1
- comparison test 1
- root test 1
- ratio test 1
- Abel Partial Summation Formula 1
- Dirichlet Test 1
- Absolute convergence 1
- Rearrangement of series 1
- Sequence of functions 1
- Series of functions 1
- Pointwise convergence 1
- Uniform convergence 1
- Cauchy criterion 1
- Weierstrass M-Test 1
- Dini Theorem 1
- integration 1
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- Analytic function 1
- basis 1
- product topology 1
- subspace 1
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- dimension theorem 1
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- Lebesgue integral 1
- Lebesgue monotone convergence theorem 1
- pseudo-inverse - linear transformation 1
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- Convolution 1
- Good Kernels 1
- Fejer's Theorem 1
- Fejer Kernel 1
- Dirichlet Problem 1
- Polar Coordinates 1
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- Infinite dimensional vector space 1
- Parseval's identity 1
- Polarized Parseval's identity 1
- Riemann-Lebesgue Lemma 1
- Localization principle of Riemann 1
- Sawtooth Function 1
- Divergent Fourier series of continuous function 1
- Curve 1
- Isoperimetric Inequality 1
- Number theory 1
- Weyl's Equidistribution theorem 1
- continuous 1
- non differentiable 1
- moderate decrease 1
- Fourier transform 1
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- Inverse Fourier transform 1
- Plancherel's theorem 1
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
moderate decrease
Fourier Transform
Definition We define (if the limit exists)
Fourier transform
Fourier Transform
Definition We define (if the limit exists)
rapidly decreasing
Fourier Transform
Definition We define (if the limit exists)
Fourier inversion
Fourier Inversion Formula
Proposition (Multiplication formula)
Inverse Fourier transform
Fourier Inversion Formula
Proposition (Multiplication formula)
Plancherel's theorem
Fourier Inversion Formula
Proposition (Multiplication formula)