Chapter 1 Glossary

This page contains a list of terms used in the course, and will be available for use on the WPR and TEE.

## Guidelines:

- Each student is required to add a minimum of two entries to each glossary.
- Entries should be
*in your own words*and not copy/pasted from elsewhere, or typed verbatim from the text. - Proper mathematical formatting using TeX is required.
- Include any terms which
*"move the story along"*, or the ones that seem to show up again and again.

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## Terms

**Cartesian product**- Given sets A and B, the Cartesian product of A and B is the set A x B of all ordered pairs (a, b) such that a $\in$ A and b $\in$ B.
**Cauchy Convergence Criterion for Sequences**- A sequence ($a_n$) in $\mathbb{R}$ converges iff $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ $\mathbb{N}$ such that n, m $\geq$ N $\Rightarrow$ |$a_n - a_m$| < $\varepsilon$.
**Complete**- There are two definitions. First: The set $\mathbb{R}$, constructed by means of Dedekind cuts, is complete if it satisfies the Least Upper Bound Property. Second, $\mathbb{R}$ is complete with respect to Cauchy sequences in the sense that if ($a_n$) is a sequence of real numbers obeying a Cauchy condition then it converges to a limit in $\mathbb{R}$.
**Continuous function**- The function $f : [a,b] \to \mathbb{R}$ is continuous if for each $\varepsilon$ > 0 and each $x \in [a, b]$ there is a $\delta$ > 0 such that $t \in [a, b]$ and $|t - x| < \delta \implies |f(t) - f(x)| < \varepsilon$.
**Convergence to a limit**- The sequence ($a_n$) converges to the limit b$\in$ $\mathbb{R}$ as $n\to\infty$ provided that for each $\epsilon$>0 there exists N $\in$ $\mathbb{N}$ such that for all n$\geq$N, $|a_n-b|$<$\epsilon$
**Convex subset**- A set $S\subset\mathbb{R}$ is convex if for any two points $x,y\in$$S$, then $tx + (1-t)y\in$$S$ for all $t\in$[0,1].
**Cut (or Dedekind cut)**- A Cut in the rationals is a pair of subsets such that

- $\ A\cup B$=$\mathbb{Q}$, where A,B are not the empty set
- $\ A\cap B$=$\o$
- If a$\in$A and b$\in$B then a<b
- A contains no largest element

**Disjoint**- If $A\cap B$ is the empty set, then
**A**and**B**are said to be disjoint. **Equivalence relation**- A relation between elements of a common set which satisfies the following three properties for $x,y,z\in S$

- $x\sim x$
- $x\sim y$ implies that $y\sim x$
- $x\sim y\sim z$ implies that $x\sim z$

**Field**- A field is a set of elements in which the two operations of addition and multiplication have the algebraic properties of being well-defined, natural, commutative, associative, and have additive and multiplicative inverses, respectively.
**Greatest lower bound (infimum)**- The most negative value that a set approaches; the highest value of everything less than the particular set.
**Injection**- If for each pair of distinct elements a,a'$\in$A, the elements f(a), f(a') are distinct in B. That is, a$\neq$a' $\Rightarrow$ f(a)$\neq$f(a').
**Irrational number**- Any number that cannot be expressed in the form a/b, where a,b$\in \mathbb{Z}$
**Least upper bound (supremum)**- The most positive value that a set approaches; the lowest value of everything greater than the particular set.
**Least upper bound property**- A nonempty subset in $\mathbb{R}$ that is bounded above has a least upper bound in $\mathbb{R}$.
**Inner product**- Also known as the
**Dot Product**of two vectors. Can be written as $\langle x,y\rangle= x_1y_1+\centerdot\centerdot\centerdot +x_my_m$. $\langle x,y\rangle$ It is a field of scalars in $\mathbb{R}$. Using the**Cauchy-Schwarz inequality**for elements of $x,y$, we can say that: $|\langle x,y \rangle|\leq ||x||\centerdot ||y||$ with equality if and only if $x$ and $y$ are linearly dependent. **Magnitude**- absolute value, or the length of a vector. Can be defined as: $|x|=\sqrt{\langle x,x\rangle}=\sqrt{x_1^2+\centerdot\centerdot\centerdot +x_m^2}$.
**Norm**- A norm on a vector space V is any function ll ll : V $\to$ $\mathbb{R}$ with the three properties of vector length, if v,w$\in$V and $\lambda$$\in$ $\mathbb{R}$ then:

a) ll v ll $\geq$ 0 and ll v ll = 0 iff v = 0.

b) ll $\lambda$v ll = l $\lambda$ l ll v ll.

c) ll v+w ll $\leq$ ll v ll + ll w ll.

**Rational numbers**- denoted as $\mathbb{Q}$; a fraction of integers in which the denominator is $\neq 0$.
**Surjection**- f: A$\to$B is a surjection if for each b$\in$B there is at least one a$\in}$A such that f(a)=b.
**Transitivity**- x<y<z implies x<z.
**Translation**- x<y implies x+z<y+z
**Trichotomy**- This is an additional property of the cut order. It states: either $x<y$, $y<x$, or $x=y$, but only one of the three things is true.
**Unit ball**- A set containing all elements with a magnitude of less than one or equal to one, $S=\{\vec{s}\in\mathbb{R}^3 | \: |\vec{s}|\leq 1\}$.

## Theorems

Unless otherwise stated, you are free to appeal to these results on the exam.

**Cauchy-Schwarz inequality**- For all $x, y \in \mathbb{R}^m, \langle x, y \rangle \leq|x| |y|$.
**Triangle inequality**- For all $x, y \in \mathbb{R}$ $|x + y| \leq|x| + |y|$.

- Also, $|x + y|^2 \leq |x|^2 + 2|x||y| + |y|^2 = (|x|+|y|)^2$

**The $\epsilon$ principle**- This is
**Theorem 8**in the book. There are 2 parts. First: If $a,b \in\mathbb{R}$ and if for each $\epsilon >0, a \leq b+\epsilon$, then $a \leq b$. Second: If $x,y \in\mathbb{R}$ and for each $\epsilon >0, |x-y| \leq\epsilon$, then $x=y$. **The "interval" theorem**(Theorem 7 in the text)- Every interval $(a,b)$, no matter how small, contains both rational and irrational numbers.
**The intermediate value theorem**- A continuous function $f$ defined on an interval $[a,b]$ takes on absolute minimum and absolute maximum values: for some $x_0, x_1 \in [a,b]$ and for all $x \in [a,b],$ $f(x_0) \leq f(x) \leq f(x_1).$
**Fundamental Theorem of Continuous Functions**- Every continuous real valued function of a real variable x$\in$[a,b] is bounded, achieves minimum, intermediate, and maximum values, and is uniformly continuous.

## General

**To show a function is continuous you have to …**- Start with "Given x an element of the domain, let $\varepsilon$ > 0." Then, solve |f(x) - f(y)| < $\varepsilon$ for |x - y|, which determines $\delta$ in terms of $\varepsilon$. Then, state "Let y be chosen such that |x - y| < $\delta$ (substitute the $\delta$ you calculated)." Finally, solve it for $\varepsilon$ to prove that |f(x) - f(y)| < $\varepsilon$.
**To show a function is***not*continuous you have to …- Show that an $\epsilon$>0 exists such that for all $\delta$ >0 there exists a u with |x-u|< $\delta$ but |f(x)-f(u)|>$\epsilon$
**To show a set is convex you have to …**- Show that any convex combination of the vectors $\mathbf{w_1}$,…, $\mathbf{w_k}$$\in$$\mathbb{R}$, $\mathbf{w}$=$s_1$$\mathbf{x_1}$+…+$s_k$$\mathbf{x_k}$, such that $s_1+\cdots+s_k$=1 and 0$\leq$$s_1$,…$\leq$$s_k$$\leq$1 is in the set.
**To show a set is***not*convex you have to …- Show there exists a convex combination (as described above) such that this convex combination is not an element of the set.