# Posts by Tag

## 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 $\mathfrak{T}$ (...

## 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 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 $N_{\epsilon} (p) \subset E$. ...

## 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

## 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 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 $N_{\epsilon} (p) \subset E$. ...

Lemma 1

## Compact set

Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded

## 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...

## 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 $a,b\in I$...

## 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, 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 and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## 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

## 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

## 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

## 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

## 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

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For all $x,y\in \mathbb{R}, \lvert x+y\rvert\leq \vert x\rvert+\vert y\rvert$

Lemma 1

Lemma 1

## 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 ...

## 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

Definition 5.2.1 Let $E\subset \mathbb{R}$ be a set and let $f:E\to\mathbb{R}$.

## 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 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 $a,b\in I$...

## 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

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 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 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 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...

## 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 $\varphi:[c,d]\to\mathbb{R}$ ...

## 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 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...

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...