# Posts by Category

## 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...

## 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 $1\leq p \leq q$,

## 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 ...

Lemma 1

## Sequences of real numbers

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

## 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.

## 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...

## 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.