midterm2 concepts

7/28/2019 Midterm2 Concepts

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Math 104theorems and concepts

for midterm 2 prep

OpenA subset U of a metric space (X, d) is open in X if for any x U r > 0 such that Br(x) U.

ClosedA subset A X is closed in X if whenever (an) is a sequence of points in A converging to somea X, then a A.

Little Side FactIf a subset U of a metric space (X, d) is open in X and S X, then US is open in S. (mentionedin lecture 10/12)

Collections of Open SetsThe union of any collection of open sets in X is open in X; the intersection of finitely many opensets in X is open in X.

Collections of Closed SetsThe intersection of any collection of closed sets in X is closed in x; the union offinitely many closedsets in X is closed in X. (proof: lecture 9/21)

Let S be a subset of a metric space X.

InteriorA point s S is said to be an interior point of S if there exists r > 0 such that Br(s) S. The set

of all interior points of S is called the interior of S in X and is denoted by intS. It is always truethat intS S. If additionally S intS (which means S = intS), then S is open.

ClosureA point x X is said to be a limit point of S if there exists a sequence (xn) S converging to x.The set of all limit points of S is called the closure of S in X and is denoted S. It is always truethat S S. If additionally S S (which means S = S), then S is closed.

DenseS is dense in X:1) if its closure in X is all of X,2) iff every nonempty open set in X contains an element of S.

BoundaryA point x X is a boundary point ofS if for any r > 0, the ball Br(p) of radius r around p containsan element ofS and an element ofX\S. The set of all boundary points ofS is called the boundaryof S and is denoted by S.


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Fun Boundary/Interior/Closure Relations1) S is open in X iff S S = .2) S is closed in X iff S S.3) For any S, S S, in other words any boundary point of S is also a limit point of S.(proof:Met Notes pg. 20 Proposition 13)

4) intS is open in X (proof: Met Notes pg. 18, Proposition 8) and S is closed in X (Homework 5problem 3).5) intS is the union of all open subsets of X contained in S, and S is the intersection of all closedsubsets of X containing S.6) S = S X\S7) S = intS S

Theorem 10.2, RossAll bounded monotone sequences converge.

Theorem 11.3, Ross

Every sequence (sn) has a monotonic subsequence. (proof: pg. 67(Ross))

Theorem 11.5 (Bolzano - Weierstrass), RossEvery bounded sequence has a convergent subsequence. (proof: pg. 69(Ross) and lecture 10/1)

ConnectednessA metric space X is disconnected if there exists nonempty, disjoint open subsets U, V X suchthat X = U V. We say that X is connected if it is not disconnected. Also, if X is connected, ifwe break it up into two disjoint open subsets U and V, then one of them is empty. (lecture 10/7)

Connected Corollary(a, b) is connected. (proof: Met Notes pg. 21 and lecture 10/7)

(Sequentially) CompactLet K (X, d). We say K is compact if every sequence in K has a convergent subsequence in K.

Compact Corollary/ Extras[a, b] is compact. (proof: lecture 10/1, Met Notes pg. 23). Using the same proof, any closed andbounded subset ofR is compact. Any finite subset {x1, . . . , xn} of a metric space X is compact.(proof: Met Notes pg. 22, ex. 28).

Heine - Borel Theorem

A subset ofRn

is compact iff it is closed and bounded. Basically we are saying that the compactsubsets ofRn are the closed and bounded subsets (note: Rn itself is not compact).

Definition of Open CoverLet X be a metric space and S X a subspace. An open cover of S is a collection {U} of opensubsets of X such that S is contained in their unionmeaning that any element of S is in at leastone of the U. A subcover of an open cover {U} is an open cover {V} of S so that each V occursin the collection {U}. An open cover is finite if it contains finitely many sets.


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(Covering) CompactA subset K of a metric space (X, d) is said to be compact if every open cover of K has a finitesubcover.

Compact Subsets Proposition

A compact subset K of a metric space X is closed and bounded in X. (2 DIFFERENT proofs:Met Notes pg. 23 Prop. 14 (sequential proof) and Met Notes pg. 25 Prop. 15 (covering proof))

ContinuityDefinition 1: A function f : M N is continuous at p M if whenever (pn) p in M, f(pn) f(p) in N.Definition 2: A function f : M N is continuous at p M iff >, > 0 such that

dN (f(q), f(p)) < whenever dM(q, p) <

Uniform ContinuityA function f : M N is uniformly continuous if > 0, > 0 such that

dN (f(q), f(p)) < whenever dM(q, p) <

Note: it is the same as continuous, except now p isnt fixed!! (Neither is q, but its not fixed inregular continuity either)

TheoremA function f : M N is uniformly continuous iff whenever dM(pn, qn) 0, then dN(f(pn), f(qn)) 0. (as stated in lecture 10/17). Iff is uniformly continuous on a set S and (sn) is a Cauchy sequence

in S, then (f(sn)) is a Cauchy sequence. (Ross pg. 138, Thm. 19.4 (proof following))

TheoremIf f : M N is continuous and M is compact, then f is uniformly continuous. (as stated inlecture 10/17, along with sketches of 2 different proofs). If f is continuous on a closed interval[a, b], then f is uniformly continuous on [a, b] (Ross pg. 136, Thm. 19.2. Same thing as said inclass, except here M = [a, b] (which is compact) and N = R.)

Two Important Corollaries:

Intermediate Value Theorem

Any continuous function f : M R where M is connected has the intermediate value property:if p, q M and f(p) < f(q), then for any z R such that f(p) < z < f(q), x M such thatf(x) = z. (proof: Met Notes pg. 30, pg.127 (Ross, Thm. 18.2), lecture 10/12)

Extreme Value TheoremAny continuous function f : M R with compact domain M achieves a maximum and a minimum.(In particular, real-valued continuous functions on compact domains are always bounded.) (proof:Met notes pg. 31, pg. 126 (Ross, Thm. 18.1), lecture 10/14)


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Other Continuity Theorems:

Theorem 12 (Metric Notes)A function f : M N is continuous iff f1(U) is open in M for any open subset U of N.

Theorem 13 (Metric Notes)The image of a connected space under a continuous function is connected, i.e. if f : M N iscontinuous and M connected, then f(S) is connected. (proof: Met Notes pg. 30)

Theorem 14 (Metric Notes)The image of a compact space under a continuous function is compact; i.e. if f : M N iscontinuous and M is compact, then f(M) is compact. (proofs: Met Notes pg. 30 (proof 1:sequential definition of compactness) and 31 (proof 2: covering definition of compactness))

Theorem 15 (Metric Notes)A continuous function on a compact metric space is uniformly continuous.

Limits

Definition of LimitLet f : (a, b) R be a function and fix x0 (a, b). We say L is the limit of f as x approaches x0if > 0, > 0 such that if 0 < |x x0| < then |f(x) L| < . Notation for limit:

limxx0

f(x) = L

limits are unique (can use 2

- trick to show this)

limxx0 f(x) = f(x0) iff f is continuous at x0

Squeeze TheoremIf f(x) g(x) h(x) x (a, b) and lim

xx0f(x) = L = lim

xx0h(x), then lim

xx0g(x) = L.

(proof: lecture 10/19)

Pointwise ConvergenceWe say (fn) converges pointwise to f : [a, b] R if x [a, b], (fn(x)) f(x). We call f thepointwise limit of (fn).

Uniform ConvergenceWe say (fn) converges uniformly to f if > 0 N N such that |fn(x) f(x)| < for n Nand x. We call f the uniform limit of (fn).

Boundedness and Uniform ConvergenceIf (fn) are bounded and (fn) f uniformly, then f is bounded. (prove using reverse triangleinequality and definition of convergence... mentioned lecture 10/26)


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Series

A series

n=1

an converges if the sequence of partial sums sn = a1 + a2 + + an converges.

A series

an converges absolutely if

|an| converges. If

an converges but

|an| doesnt, call

this conditional convergence.

If

an is absolutely convergent, rearrangements dont affect the convergence.

TheoremSuppose

an is conditionally convergent. Then given any x R, a rearrangement of

an

converging to x.

Cauchy Criterion for Convergencean satisfies the Cauchy criterion if its sequence of partial sums (sn) is a Cauchy sequence, in

other words, for each > 0, N N such that n m > N implies

n

k=m

ak

< . A series converges

iff it satisfies the Cauchy criterion.

TheoremIf a series

an converges, then lim an = 0. (proof: pg. 90 (Ross), Corollary 14.5)

Series of FunctionsA series of functions

fn converges (pointwise or uniformly) if the sequence of partial sums

gk = f1 + f2 + + fk converges (pointwise or uniformly)

Types of SeriesGeometric: of the form

n=0 arn. For r = 1, given by

n

k=0

ark = a1 rn+1

1 r

1

1 aif|a| < 1, diverges else

( pg. 91 (Ross))

Power: of functions of the form

n=0

an(x x0)n

xn on R doesnt converge on R, but converges pointwise (NOT uniformly!) on (-1, 1) to 11x

.

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