Metric Spaces

Question 1

Define differentiable in a metric space

Answer 1

We say that a function f: Ω ⊆ Rn → Rm is differentiable at a ∈ Ω if there exists a linear map L: Rn → Rm such that lim(h→0) [f(a + h) − f(a)]/|h| = L.

Question 2

Define a directional derivative

Answer 2

We say that a function f : Ω ⊆ Rn → Rm is differentiable at a ∈ Ω in direction n if lim(λ→0)[f(a+λn)−f(a)]/λ exists.

Question 3

Define a path

Answer 3

Let U ⊆ R2 and a0, a1 ∈ U. A path in U between a0 and a1 is a continuous map γ : [0, 1] → U with γ(0) = a0 and γ(1) = a1.

Question 4

Define path connected

Answer 4

U is path connected if for every a0, a1 ∈ U there exists a path between a0 and a1.

Question 5

Define homotopy

Answer 5

Let U ⊆ R2 and a0, a1 ∈ U. Let γ0 and γ1 be paths in U connecting a0 and a1. We say that the paths γ0 and γ1 are homotopic if there exists a continuous map Γ : [0, 1] × [0, 1] → U with Γ(s, 0) = a0 and Γ(s, 1) = a1 for all s ∈ [0, 1], and Γ(0, t) = γ0(t) and Γ(1, t) = γ1(t) for all t ∈ [0, 1].

Question 6

Define simply connected

Answer 6

U ⊆ R2 is simply connected if it is path-connected and given any points a0, a1 ∈ U then any two paths in U between a0 and a1 are homotopic.

Question 7

Define a distance function

Answer 7

Let X be a set. Then d: X x X → R is a distance function if it possesses:
Positivity: d(x, y) ≥0 and d(x, y) = 0 if and only if x = y
Symmetry: d(x, y) = d(y, x)
Triangle inequality: if x, y, z ∈ X then we have d(x, z) ≤ d(x, y) + d(y, z).

Question 8

Define a metric space

Answer 8

A set coupled with a distance function.

Question 9

Give the reverse triangle inequality

Answer 9

Let x, y, z be points in a metric space. Then we have |d(x, y) − d(x, z)| ≤ d(y, z).

Question 10

Define the d1 metric

Answer 10

If X = Rn, d1(v, w) = Sum from 1 to n of |v_i – w_i|.

Question 11

Define the d2 metric

Answer 11

If X = Rn, d2(v, w) = sqrt(Sum from 1 to n of (v_i – w_i)^2).

Question 12

Define the d∞ metric

Answer 12

If X = Rn, d∞(v, w) = max of |v_i – w_i|

Question 13

Define the discrete metric

Answer 13

Let X be any set, then the discrete metric, d(x,y) = 1 if x ≠ y and d(x,y) = 0 otherwise.

Question 14

Define the p-adic metric

Answer 14

Let X = Z, and the p-adic metric, d(x,y) = p^(−m), where p^m is the largest power of two dividing x – y.

Question 15

Define a norm

Answer 15

Let V be a vector space over R. A function ∥·∥: V → [0,∞) is called a norm if:
∥x∥ = 0 if and only if x = 0
∥λx∥ = |λ|∥x∥ for all λ ∈ R, x ∈ V
∥x + y∥ ≤ ∥x∥ + ∥y∥ whenever x, y ∈ V.

Question 16

Define a subspace of a metric space

Answer 16

Suppose (X,d) is a metric space, then for any subset Y of X, (Y,d) is a subspace of (X,d)

Question 17

Define the product space of metric spaces

Answer 17

If (X, dX) and (Y, dY) are metric spaces, then if we let d(XxY)((x1, y1),(x2, y2)) = sqrt(dX(x1, x2)^2 + dY(y1, y2)^2, then (XxY,d(XxY) is the product space.

Question 18

Define an open ball

Answer 18

Let X be a metric space a ∈ X and ε>0. Then the open ball about a of radius ε is the set B(a, ε) = {x ∈ X : d(x, a) < ε}.

Question 19

Define a closed ball

Answer 19

Let X be a metric space a ∈ X and ε>0. Then the closed ball about a of radius ε is the set B(a, ε) = {x ∈ X : d(x, a) ≤ ε}.

Question 20

Define a bounded set

Answer 20

Let X be a metric space, and let Y ⊆ X. Then we say that Y is bounded if Y is contained in some open ball.

Question 21

Define a limit of a sequence

Answer 21

Suppose that (x_n) is a sequence of elements of a metric space (X, d) and x ∈ X. Then we say that x_n → x, if for every ε > 0, there is an N such that d(x_n, x) < ε for all n ≥ N.

Question 22

Define continuity

Answer 22

Let (X, dX) and (Y, dY) be metric spaces. We say a function f: X → Y is continuous at a ∈ X if for any ε > 0 there is a δ > 0 such that for any x ∈ X, dX(a, x) < δ implies that dY(f(x), f(a)) < ε. We say f is continuous if it is continuous at every a ∈ X.

Question 23

Define uniform continuity

Answer 23

Let (X, dX) and (Y, dY ) be metric spaces. We say a function f X → Y is uniformly continuous if for any ε > 0 there is a δ > 0 such that for any x, y ∈ X with dX(x, y) < δ we have dY(f(x), f(y)) < ε.

Question 24

Define the bounded function space

Answer 24

If X is any set, we define B(X) to be the space of functions f: X → R for which f(X) = {f(x) : x ∈ X} is bounded.

Question 25

Define the continuous function space

Answer 25

If X is a metric space, we define C(X) to be the space of functions f: X → R for which f(X) = {f(x) : x ∈ X} is continuous.

Question 26

Define the continuous bounded function space

Answer 26

If X is a metric space, C_b(X) = C(X) ∩ B(X).

Question 27

Define an isometry

Answer 27

Let (X, dX) and (Y, dY) be metric spaces. A function f: X → Y between metric spaces (X, dX) and (Y, dY) is said to be an isometry if dY(f(x), f(y)) = dX(x, y) for all x, y ∈ X.

Question 28

Define a bijective isometry

Answer 28

A surjective isometry (as by definition it is already injective).

Question 28

Define the bijective isometry group

Answer 28

Isom(X) is the set of all bijective isometries under composition.

Question 29

Define a homeomorphism

Answer 29

Let X and Y be metric spaces. Then f is a homeomorphism if it is continuous, a bijection, and if its inverse f^(−1): Y → X is also continuous.

Question 30

Define an open set

Answer 30

Let X be a metric space. Then a subset U ⊆ X is open if for each u ∈ U there is some δ > 0 such that the open ball B(u,δ) is contained in U.

Question 31

Define a closed set

Answer 31

Let X be a metric space. Then a subset U ⊆ X is closed if its complement, X\U is an open subset of X.

Question 32

Define an interior

Answer 32

Let X be a metric space, and let S ⊆ X. The interior of S is defined to be the union of all open subsets of X contained in S.

Question 33

Define closure

Answer 33

Let X be a metric space, and let S ⊆ X. The closure of S is defined to be the intersection of all closed subsets of X containing S.

Question 34

Define a boundary

Answer 34

Let X be a metric space, and let S ⊆ X. ∂S = closure(S)\interior(S).

Question 35

Define dense

Answer 35

Let X be a metric space, and let S ⊆ X. S is said to be dense in X if the closure of S is X.

Question 36

Define a limit point

Answer 36

If X is a metric space and S ⊆ X is a subset, then a ∈ X is a limit point of S if any open ball about a contains a point of S other than a itself.

Question 37

Define an isolated point

Answer 37

If X is a metric space and S ⊆ X is a subset, then a is an isolated point of S if a ∈ S but a is not a limit point.

Question 38

Give the alternative definition of a closed set

Answer 38

Let S be a subset of a metric space X. Then S is closed if and only if it contains all its limit points.

Question 39

Define a bounded sequence

Answer 39

Let (xn) be a sequence in some metric space X. Then (xn) is bounded if there exists a and R such that the set {xn : n ≥ 1} all lie in some open ball B(a, R).

Question 40

Define a Cauchy sequence

Answer 40

Let (xn) be a sequence in some metric space X. Then (xn) is Cauchy if for every ε > 0, there is some N such that d(xn, xm) < ε whenever n, m ≥ N.

Question 41

Define a convergent sequence

Answer 41

Let (xn) be a sequence in some metric space X. Then (xn) is convergent if there is some a ∈ X such that lim(n→∞)xn = a.

Question 42

Define completeness

Answer 42

A metric space is complete if every Cauchy sequence converges.

Question 43

Define diameter

Answer 43

Let X be a metric space and Y ⊆ X a non-empty subset. The diameter of Y is the supremum of the set {d(x, y) : x, y ∈ Y } when this set is bounded, and infinity otherwise.

Question 44

State Cantor's intersection theorem

Answer 44

Let X be a complete metric space and suppose that S1 ⊇ S2 ⊇ . . . form a nested sequence of non-empty closed sets in X with the property that diam(Sn) → 0 as n → ∞. Then the intersection of Sn contains a unique point a.

Question 45

Define Lipschitz

Answer 45

Let (X, dX) and (Y, dY) be metric spaces and suppose that f: X → Y. We say that f is a Lipschitz map if there is a constant K ≥ 0 such that dY(f(x), f(y)) ≤ KdX(x, y).

Question 46

Define a contraction

Answer 46

Let (X, dX) and (Y, dY) be metric spaces and suppose that f: X → Y. If Y = X and f is Lipschitz with Lipschitz constant K ∈ [0, 1) then we say that f is a contraction.

Question 47

State the contraction mapping theorem

Answer 47

Let X be a nonempty complete metric space and suppose that f: X → X is a contraction. Then f has a unique x ∈ X such that f(x) = x.

Question 48

Define a connected metric space

Answer 48

A metric space is connected if it cannot be written as the disjoint union of two nonempty open sets.

Question 49

Define disconnecting sets

Answer 49

If X is a metric space that can be written as a disjoint union of two nonempty open sets U and V then we say that these sets disconnect X.

Question 50

Give the sunflower lemma

Answer 50

Let X be a metric space and {Ai: i ∈ I} be a collection of connected subsets of X such that the intersection of all Ai is non-empty. Then the union of all Ai is connected.

Question 51

Define a path connected metric space

Answer 51

Let X be a metric space. Then we say that X is path-connected if for any a, b ∈ X there exists a path from a to b.

Question 52

Define path components

Answer 52

Let X be a metric space and define the equivalence relation ∼ on X as follows: a ∼ b if and only if there is a path connecting a to b. The equivalence classes into which this relation partitions X are called the path-components of X.

Question 53

Define sequential compactness

Answer 53

Let X be a metric space. Then X is said to be sequentially compact if any sequence of elements in X has a convergent subsequence.

Question 54

State Bolzano Weierstrass

Answer 54

Any closed and bounded subset of Rn is sequentially compact.

Question 55

Define totally bounded

Answer 55

A metric space is totally bounded if, for any ε > 0, it may be covered by finitely many open balls of radius ε.

Question 56

Define an open cover

Answer 56

Let X be a metric space and U = {Ui: i∈I} a collection of open subsets of X. We say that U is an open cover of X if X = the union of Ui, where i∈I.

Question 57

Define a subcover

Answer 57

Let X be a metric space and U = {Ui: i∈I} by an open cover. If J ⊆ I is a subset such that X = the union of Ui, where i∈J then we say that {Ui: i ∈ J} is a subcover of U.

Question 58

Define compactness

Answer 58

A metric space is said to be compact if every open cover has a finite subcover.

Question 59

Define a Mobius map

Answer 59

Each element g ∈ GL2(C) gives a Mobius map Ψg: C∞ → C∞. Roughly, this is given by the formula Ψg(z) = (az + b)/(cz + d), but one needs to be careful about ∞. If c ≠ 0 then we define Ψg(−d/c) = ∞ and Ψg(∞) = a/c, while if c = 0 then we define Ψg(∞) = ∞.

Question 60

Define a circline

Answer 60

A circline is either a circle in C or a line in C together with the point {∞}.

Question 61

Define a tangent line of a path

Answer 61

If γ: [0, 1] → C is a C1 path which has γ’(t) ≠ 0 for all t, then {γ(t) + sγ′ (t) : s ∈ R} is the tangent line to γ at γ(t).

Question 62

Define a tangent vector of a path

Answer 62

If γ: [0, 1] → C is a C1 path which has γ’(t) ≠ 0 for all t, then γ’(t) is a tangent vector at γ(t) ∈ C.

Question 63

Define conformal

Answer 63

Let U be an open subset of C and suppose that T: U → C is continuously differentiable in the real sense. T is conformal at z0 if for every pair of C1 paths γ1, γ2 through z0, the angle between their tangent vectors at z0 is equal to the angle between the tangent vectors at T(z0) given by the C1 paths T◦γ1 and T◦γ2. We say that T is conformal on U if it is conformal at every z ∈ U.