Metric Spaces Flashcards
Define differentiable in a metric space
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.
Define a directional derivative
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.
Define a path
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.
Define path connected
U is path connected if for every a0, a1 ∈ U there exists a path between a0 and a1.
Define homotopy
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].
Define simply connected
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.
Define a distance function
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).
Define a metric space
A set coupled with a distance function.
Give the reverse triangle inequality
Let x, y, z be points in a metric space. Then we have |d(x, y) − d(x, z)| ≤ d(y, z).
Define the d1 metric
If X = Rn, d1(v, w) = Sum from 1 to n of |v_i – w_i|.
Define the d2 metric
If X = Rn, d2(v, w) = sqrt(Sum from 1 to n of (v_i – w_i)^2).
Define the d∞ metric
If X = Rn, d∞(v, w) = max of |v_i – w_i|
Define the discrete metric
Let X be any set, then the discrete metric, d(x,y) = 1 if x ≠ y and d(x,y) = 0 otherwise.
Define the p-adic metric
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.
Define a norm
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.
Define a subspace of a metric space
Suppose (X,d) is a metric space, then for any subset Y of X, (Y,d) is a subspace of (X,d)
Define the product space of metric spaces
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.
Define an open ball
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}.
Define a closed ball
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}.
Define a bounded set
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.
Define a limit of a sequence
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.
Define continuity
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.
Define uniform continuity
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.
Define the bounded function space
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.
Define the continuous function space
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.
Define the continuous bounded function space
If X is a metric space, C_b(X) = C(X) ∩ B(X).
Define an isometry
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.
Define a bijective isometry
A surjective isometry (as by definition it is already injective).
Define the bijective isometry group
Isom(X) is the set of all bijective isometries under composition.
Define a homeomorphism
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.
Define an open set
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.
Define a closed set
Let X be a metric space. Then a subset U ⊆ X is closed if its complement, X\U is an open subset of X.
Define an interior
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.
Define closure
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.
Define a boundary
Let X be a metric space, and let S ⊆ X. ∂S = closure(S)\interior(S).
Define dense
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.
Define a limit point
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.
Define an isolated point
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.
Give the alternative definition of a closed set
Let S be a subset of a metric space X. Then S is closed if and only if it contains all its limit points.
Define a bounded sequence
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).
Define a Cauchy sequence
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.
Define a convergent sequence
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.
Define completeness
A metric space is complete if every Cauchy sequence converges.
Define diameter
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.
State Cantor’s intersection theorem
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.
Define Lipschitz
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).
Define a contraction
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.
State the contraction mapping theorem
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.
Define a connected metric space
A metric space is connected if it cannot be written as the disjoint union of two nonempty open sets.
Define disconnecting sets
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.
Give the sunflower lemma
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.
Define a path connected metric space
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.
Define path components
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.
Define sequential compactness
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.
State Bolzano Weierstrass
Any closed and bounded subset of Rn is sequentially compact.
Define totally bounded
A metric space is totally bounded if, for any ε > 0, it may be covered by finitely many open balls of radius ε.
Define an open cover
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.
Define a subcover
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.
Define compactness
A metric space is said to be compact if every open cover has a finite subcover.
Define a Mobius map
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(∞) = ∞.
Define a circline
A circline is either a circle in C or a line in C together with the point {∞}.
Define a tangent line of a path
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).
Define a tangent vector of a path
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.
Define conformal
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.
