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- Wisconsin
- University of Wisconsin - Madison
- Mathematics
- Mathematics 521
- Stovall
- Analysis Final

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Definition of one-to-one

Let A and B be two non-empty sets and f: A --> B a function. We say that f is one-to-one if for every x1, x2 in A, if f(x1) = f(x2), then x1 = x2

Well-ordering Principle

Every non-empty subset of N has a unique least element

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Definition of a Field

A field is a set F, containing at least two elements, and equipped with two binary operations, + and *, that satisfy the following axioms:

1) (+ closure), + commutes, + associative, + identity, + inverse

2) multiplicative closure, commute, associative, identity, inverse, distributivity

1) (+ closure), + commutes, + associative, + identity, + inverse

2) multiplicative closure, commute, associative, identity, inverse, distributivity

+ closure

for any x, y in F, x + y is in F

+ commute

for any x, y in F, x + y = y + x

+ associative

for any x, y, z in F, (x + y) + z = x + (y+z)

+ identity

there exists a 0 in F such that x + 0 = x for any x in F

Ordered Set

Let S be a set. An order on S is a binary relation satisfying the following axioms

1) for any x,y in S, exactly one of the following holds: x=y, x<y, y<x

2) Transitivity: for any x, y, z in S, if x < y and

y<z, then x <z

1) for any x,y in S, exactly one of the following holds: x=y, x<y, y<x

2) Transitivity: for any x, y, z in S, if x < y and

y<z, then x <z

+ inverse

for any x in F there exits (-x) in F such that x + (-x) = 0

multiplicative closure

for any x,y in F, x * y is in F

multiplicative commutes

for any x, y in F x * y = y * x

multiplicative associative

for any x,y,z in F (x * y) * z = x * (y *z)

Multiplicative identity

there exits a 1 in F with 1 not equal to 0 such that for any x in F, 1 * x = x

multiplicative inverse

for any x in F, x does not equal to 0, there exists a 1/x in F such that x * 1/x = 1

Distributivity

for any x, y, z in F x * (y+z) = x*y + x*z

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Ordered Field

An ordered Field is a field F, which is also an ordered set, such that the following hold for any x,y,z in F

1) if y is less than or equal to z, then x + y is less than or equal to x +z

2) if x is greater than or equal to 0 and y is greater than or equal to 0, then x*y is greater than or equal to 0.

1) if y is less than or equal to z, then x + y is less than or equal to x +z

2) if x is greater than or equal to 0 and y is greater than or equal to 0, then x*y is greater than or equal to 0.

Upper bound

B is an upper bound for E if and only if for any x in E, x is less than or equal to B

Lower Bound

B is a lower bound for E if and only if for x in E x is greater than or equal to B.

Supremum

B is the sup of E if and only if B is the least element of the set of upper bounds for E.

Least Upper Bound Property

- Say that the ordered set S has the least upper bound property if every nonempty, bounded above subset E contained in S has a supremum in S.

Archimedean Property

Let x,y in R with x > 0. There exits n in N(natural) such that nx > y

Countably Infinite

E is countably infinite if there exists a bijection between E and N (the natural numbers)

When is E finite?

We say E is finite if #E is {0} U the natural numbers

Let X be a nonempty set, a metric on X is a function X: X --> R such that for any x,y,z in X

i) (positivity) d(x,y) is greater than or equal to 0 and d(x,y) = 0 if and only if x = y

ii) (symmetry) d(x,y) = d(y,x)

iii) (Triangle Inequality) d(x,z) is less than or equal to d(x,y) + d(y,z)

i) (positivity) d(x,y) is greater than or equal to 0 and d(x,y) = 0 if and only if x = y

ii) (symmetry) d(x,y) = d(y,x)

iii) (Triangle Inequality) d(x,z) is less than or equal to d(x,y) + d(y,z)

Interior Point

Let E be contained in X. Then p is an interior Point of E if there exists an r > 0 such that N_{r}(p) is contained in E.

Openness

E is open if every Point of E is an interior Point

Accumulation Point

Let E be contained in X. Say p is an accumulation Point of E if for any r > 0 (N_{r}(p) ∩ E) \ {p} ≠ ∅

Isolated Point

P is an isolated point of E if p is in E \ E'

Alternative Definition for accumulation Point

Let E be contained in X and let p in X. Then p is in E' if and only if for any r > 0, N_{r}(p) ∩ E contains infinitely many points.

Closure

E is closed if E contains all of its accumulation points

Closure/openness for unions/intersections

1) arbitrary unions and finite intersections of open sets are open

2) arbitrary intersections and finite unions of closed sets are closed

2) arbitrary intersections and finite unions of closed sets are closed

Open Cover

Subcover

Compact

Heine-Borel Theorem

In ℝ^{k} with Euclidean metric, a set is compact <=> it is closed and bounded.

Separated Sets

Two sets A and B are separated if

A ≠ ∅, B ≠ ∅, Ā ∩ B = ∅ and

B̄ ∩ A = ∅

A ≠ ∅, B ≠ ∅, Ā ∩ B = ∅ and

B̄ ∩ A = ∅

Seperation

A and B form a separation of E contained in X if E = A U B, A ≠ ∅, B ≠ ∅,

Ā ∩ B = ∅ and B̄ ∩ A = ∅

E Is not connected if there exists a separation of E

Ā ∩ B = ∅ and B̄ ∩ A = ∅

E Is not connected if there exists a separation of E

Convergence of a Sequence

The Sequence (p_{n}) in X converges to p( in X) if for any Ɛ > 0 there exists N in N(natural numbers) such that for any n > N, d(p_{n},p) < Ɛ.

Squeeze Theorem

Let (x_{n}), (y_{n}), (z_{n}) be three sequences in R. Assume that Lim x_{n} = x = lim z_{n} and that for any n in N (natural numbers) x_{n} ≤ y_{n} ≤ z_{n}. Then lim y_{n} = x.

Subsequences

Subsequential limit

P is a sub-sequential limit of (p_{n}) if there exists a subsequent (p_{nk}) of (p_{n}) such that limit as k goes to infinity of p_{nk} = p.

Alternative Definition of Subsequential limit

Let (p_{n}) be a Sequence in X. Then p in X is a sub-sequential limit of (p_{n}) <=> for any Ɛ > 0, N_{Ɛ}(p) contains p_{n} for infinitely many n's.

Cauchy Sequence

We say (p_{n}) is a Cauchy Sequence if for any Ɛ > 0 there exists an N in N (natural numbers) such that for any n, m ≥ N , d(p_{n}, p_{m}) < Ɛ.

Increasing

The Sequence S_{n} in R is increasing if S_{n+1} ≥ S_{n} for any n in N(natural numbers)

Decreasing

The Sequence (S_{n}) in R is decreasing if S_{n+1} ≤ S_{n} for any n in N(natural numbers)

Monotone

S_{n} is monotone if ( for any n in N (natural numbers) S_{n+1} ≥ S_{n}) or ( for any n in N(natural numbers) S_{n+1} ≤ S_{n}.

Not Monotone

(S_{n}) is not monotone if there exists an n in N(natural numbers) such that S_{n+1} < S_{n} and if there exists an m in N(natural numbers) such that S_{m+1} > S_{m}.

Divergence to +∞

Let (S_{n}) be a Sequence in R. Say (S_{n}) diverges to +∞ if for any M > 0 there exists an N in N (natural numbers) such that for any n ≥ N, S_{n} > M

Divergence to -∞

Let (Sn) be a Sequence in R. Say (Sn) diverges to -∞ if for any M > 0 there exists an N in N (natural numbers) such that for any n ≥ N, Sn < -M

Lim sup of a Sequence

Lim inf of a Sequence

Infinite Series Convergence

Cauchy Criterion for Convergence

Absolute Convergence

We say that the the series summation of a_{n} converges absolutely if the summation of |a_{n}| converges.

Cauchy Condensation Test

Root Test for Convergencea

Ratio Test for Convergence

Summation by Parts

Conditional Convergence

Limits of functions on metric spaces

Continuity of a function

Uniformly Continous

Differentiability

Chain rule

Local Max

Intermediate Value Theorem

Rolle's Theorem

Mean Value Theorem

L'Hopital's Rule

Higher Order Derivatives

Taylor's Theorem with Remainder

Partition

Upper Riemann Sum

Lower Riemann Sun

Upper Riemann Integral

Lower Riemann Integral

Riemann Integrable

Refinement

Improper Integrals

Antiderivative

Fundamental Theorem of Calculus I

Fundamental Theorem of Calculus II

Integration by Parts

Change of Variables

Cauchy Criterion for improper integrals

Comparison test for improper integrals

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