Semester 1 Definitions Flashcards
A metric space
A metric space is a pair (X,d) where X is a set and d:X x X -> R is a map such that for all x,y,z:
1:d(x,y) >= 0
2:d(x,y) = 0 => x = y
3:d(x,y) = d(y,x)
4:d(x,y) =< d(x,z) + d(z,y)
Open Ball
For any x£X and r>0:
B(x,r) = {y £ X |d(x,y) < r}
Closed Ball
For any x £ X and r > 0:
B(x,r) = {y £ X|d(x,y) =< r}
Bounded subset
A subset is called bounded if it is contained in some open ball
The diameter of a subset
d(A) := sup d(x,y) with x,y £ A
Internal Point
A point x £ A is an internal point if there exists r > 0 such that B(x,r) c A. Where A c X
Open set
A subset is called an open set if every point of A is an internal point
Interior
The set of all internal points of A c X
Limit point
A point x £ X is called a limit point of A if for any e > 0 there exists y £ Awithout{x} such that d(x,y) < e
Closed set
A is called a closed set if it contains all of its limit points
Closure
The union of A and the set of all of its limit points.
Convergent sequence
A sequence (Xn) of points in X is called convergent to x £ X if for any e>0 there exists N>0 such that:
d(Xn,x) < e for any n >= N
Converge pointwise
A sequence (Xn(t)) is said to converge pointwise to a function x(t) if for any t £ [a,b] we have lim n-inf Xn(t) = x(t)
Uniform convergence
A sequence (Xn(t)0 is said to converge uniformly to a function x(t) if for an e>0 there exists N>0:
|Xn(t) - x(t)|<e
for any n>=N for any t £ [a,b]
Difference between pointwise and uniform convergence
(1) Note that for pointwise convergence we can choose N depending on t, while for uniform convergence
we have to choose N that works for all t simultaneously (uniformly).
(2) Uniform convergence implies pointwise convergence.
(3) We can see from the previous example that uniform convergence is equivalent to the convergence
with respect to the metric d∞.
Continuous at a point
x £ X if for any e>0 there exists 6>0:
d(x,y)<6 => d’(f(x),f(y)) < e
Continuous
Every point is continuous
Lipschitz continuous
A map f:(X,d) -> (Y,d’) is called Lipschitz continuous if there exists L>= such that:
d’(f(x),f(y)) =< L.d(x,y) for any x,y £ X
Isometry
A map f:(X,d) -> (Y,d’) between two metric spaces is called an isometry if
d’(f(x),f(y)) = d(x,y)
Global Isometry
Bijective isometry
Cauchy Sequence
A sequence Xn in X is called a Cauchy sequence if for an e>0 there exists an N>0:
d(Xm,Xn) < e for any m,n >= N
Complete metric space
A metric space is called a complete metric space if every cauchy sequence in X converges in X
Dense
A subset A c X is called dense if the closure equals the whole set. A closure = X
Completion of a metric space
A completion of a metric space (X,d) is a complete metric space (X,d) together with an isometry i:X -> X* such that the closure of i(X) = X* (i(X) is dense in X*)
Contraction
A map is called a contraction if there exists 0<a<1 such that for any x,y £ X:
d(f(x),f(y)) =< a.d(x,y)
Fixed point
An element is called a fixed point of a map f:X -> X if f(x)=x
A Topological space
Is a pair (X,T) where X is a set and T is a collection of subsets of X such that:
(1) null set, X £ T
(2) The union of a collection of sets in T is also in T
(3) The intersection of a finite collection of sets in T is in T
Metrizable
A topological space (X,T) is called metrizable if T is generated by some metric on X
Closed (Topology)
A subset A c X is called closed if its complement is open
Closure (Topology)
Given a subset A c X, its closure is the minimal closed set that contains A. It is the intersection of all the closed sets containing A
Interior (Topology)
Given a subset A c X its interior is defined to the maximal set contained in A. It is the union of all open sets contained in A
Continuous (Topology)
A map f:(X,T) -> (Y,T’) is called continuous if for every open set U c Y the preimage f-1(u) is open in X
Homeomorphism
A map that is bijective and f,f-1 are both continuous
Subspace topology
Let (X,T) be a top space and A c X be a subset. Then the collection Ta = {UnA | U £ T} is a topology on A called the subspace topology
Product topology
Let (X,T) and (Y,T’) be two topological spaces. We define the topology on X x Y to be the collection of all unions of Ui x Vi where Ui are open sets of X and Vi are open sets of Y for i £ I. This topology is called the product topology
Open Neighbourhood( topology)
An open set U c X that contains x is called an open neighbourhood of x
Neighbourhood (topology)
A subset V c X is called a neighbouhood of x if there exists an open set U c X such that x £ U c V
Convergent (topology)
Let (X,T) be a topological space. A sequence (Xn) in X is called convergent to x £ X if for any neighbourhood U with x £ U, there exists N > 0 such that Xn £ U for n >= N
Hausdorff space
A top space (X,T) is called a hausdorff space if for any x,y £ X with x =/ y there exists open sets U and V, with x £ U, y £ V, such that UnV = nullset
Connected
A top space is called connected if X can not be represented as a disjoint union X = UuV of two nonempty open subsets U,V. Equivalently we require that if U c X is both open and closed then U = nullset or U = X
Connected component
A subset A c X is called a connected component of X if it is connected and it is not contained in any larger connected subset
Path connected
A top space is called path-connected if for any two points x,y £ X there exists a continuous map f:[0,1] -> X such that f(0) = x, f(1) = y
Open cover
A collection of open sets in X is called an open cover of X if X = the union of this collection of sets
Compact topological space
X is called a compact topological space if every open cover of it contains a finite subcover
Compact
A subset A c X is called compact if (A,Ta) is a compact top space
Sequentially compact
A metric is sequentially compact if any sequence (Xn) in X has a convergent subsequence
e - net
Given e > 0, a subset A c X is called an e-net of X if for any x £ X there exists y £ A:
d(x,y) < e
Totally bounded
A metric space X is called totally bounded if for any e > 0 there exists a finite e-net of X
Normed Vector space
A pair (X,II.II) where X is a vector space and II.II is a map II.II : X -> R, x -> IIxII, satisfying:
(1) IIxII = 0 <=> x = 0
(2) IIkxII = IkI . IIxII for all k e K, x e X
(3) IIx+yII =< IIxII + IIyII for all x,y e X
Vector space
A vector space over a field K is a set X together with two operations:
Addition and scalar multiplication, satisfying:
(1) (X,+) is an abelian group
(2) k(ux) = (ku)x
(3) 1x = x
(4) k(x+y) = kx + ky
(5) (k + u)x = kx + ux
for any x,y e X, and k,u e K
Linear operator
Let (X,II.II), (Y,II.II) be two normed vector spaces over K.
A map A: X -> Y is called a linear operator if:
A(x+y) = Ax + Ay, A(kx) = kAx
for any x,y e X and k e K
Bounded linear operator
Let (X,II.II), (Y,II.II) be two normed vector spaces over K.
A linear operator A: x -> Y is called bounded if there exists M>=0 such that:
IIAxII =< M.IIxII
Continuous linear operator
Let (X,II.II), (Y,II.II) be two normed vector spaces over K.
A linear operator A: X -> Y is called continuous if it is continuous with respect to the metrics on X and Y induced by the norms
Equivalent Norms
Two norms II.II_1 and II.II_2 on a vector space X are called equivalent if there exists C,C’ > 0 such that
IIxII_2 =< C.IIxII_1, IIxII_1 =< C’.IIxII_2
A banach space
A banach space is a normed vector space (X, II.II ) which is complete
Convergent series
The formal series is called convergent to x £ X if the sequence of partial sums converges to x.
Absolute convergence
The partial sums of absolute values converges to x
Invertible
Let X be a normed vector space and let I £ L(X) be the identity operator defined by I(x) = x for x £ X.
An operator A £ L(x) is called invertible in L(x) if there exists an operator B £ L(x) such that AB = BA = I.
Homeomorphism
A map f is called a homeomorphism if f is bijective and both f,f-1 are continuous
Equicontinuous
A is equicontinuous, that is, ∀ε > 0 ∃δ > 0
∀f ∈ A:
d(x, y) < δ =⇒ |f(x) − f(y)| < ε
Uniformly continuous
A map f: X → Y between two metric spaces is called uniformly continuous if ∀ε > 0
∃δ > 0:
dX(x, y) < δ =⇒ dY (f(x), f(y)) < ε
Maximum
The maximum max A to be an element c ∈ A such that x ≤ c for all x ∈ A.
Minimum
The minimum min A to be an element c ∈ A such that x ≥ c for all x ∈ A
Supremum
The supremum sup A ∈ R ∪ {+∞, −∞} to be the least upper bound of A, that is, an element
c ∈ R ∪ {+∞, −∞} such that x ≤ c for all x ∈ A (upper bound of A) and c is the least element
with this property, meaning that if x ≤ d for all x ∈ A, then c ≤ d.
Infimum
The infimum inf A ∈ R ∪ {+∞, −∞} to be the greatest lower bound of A, that is, an element
c ∈ R∪{+∞, −∞} such that x ≥ c for all x ∈ A (lower bound of A) and c is the greatest element with this property, meaning that if x ≥ d for all x ∈ A, then c ≥ d.
