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.

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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 \lambdav 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\toB is a surjection if for each b\inB 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.