Semester 1 Definitions

Question 1

A metric space

Answer 1

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)

Question 2

Open Ball

Answer 2

For any x£X and r>0:
B(x,r) = {y £ X |d(x,y) < r}

Question 3

Closed Ball

Answer 3

For any x £ X and r > 0:
B(x,r) = {y £ X|d(x,y) =< r}

Question 4

Bounded subset

Answer 4

A subset is called bounded if it is contained in some open ball

Question 5

The diameter of a subset

Answer 5

d(A) := sup d(x,y) with x,y £ A

Question 6

Internal Point

Answer 6

A point x £ A is an internal point if there exists r > 0 such that B(x,r) c A. Where A c X

Question 7

Open set

Answer 7

A subset is called an open set if every point of A is an internal point

Question 8

Interior

Answer 8

The set of all internal points of A c X

Question 9

Limit point

Answer 9

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

Question 10

Closed set

Answer 10

A is called a closed set if it contains all of its limit points

Question 11

Closure

Answer 11

The union of A and the set of all of its limit points.

Question 12

Convergent sequence

Answer 12

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

Question 13

Converge pointwise

Answer 13

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)

Question 14

Uniform convergence

Answer 14

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]

Question 15

Difference between pointwise and uniform convergence

Answer 15

(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∞.

Question 16

Continuous at a point

Answer 16

x £ X if for any e>0 there exists 6>0:
d(x,y)<6 => d’(f(x),f(y)) < e

Question 17

Continuous

Answer 17

Every point is continuous

Question 18

Lipschitz continuous

Answer 18

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

Question 19

Isometry

Answer 19

A map f:(X,d) -> (Y,d’) between two metric spaces is called an isometry if
d’(f(x),f(y)) = d(x,y)

Question 20

Global Isometry

Answer 20

Bijective isometry

Question 21

Cauchy Sequence

Answer 21

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

Question 22

Complete metric space

Answer 22

A metric space is called a complete metric space if every cauchy sequence in X converges in X

Question 23

Dense

Answer 23

A subset A c X is called dense if the closure equals the whole set. A closure = X

Question 24

Completion of a metric space

Answer 24

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*)

Question 25

Contraction

Answer 25

A map is called a contraction if there exists 0

Question 26

Fixed point

Answer 26

An element is called a fixed point of a map f:X -> X if f(x)=x

Question 27

A Topological space

Answer 27

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

Question 28

Metrizable

Answer 28

A topological space (X,T) is called metrizable if T is generated by some metric on X

Question 29

Closed (Topology)

Answer 29

A subset A c X is called closed if its complement is open

Question 30

Closure (Topology)

Answer 30

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

Question 31

Interior (Topology)

Answer 31

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

Question 32

Continuous (Topology)

Answer 32

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

Question 33

Homeomorphism

Answer 33

A map that is bijective and f,f-1 are both continuous

Question 34

Subspace topology

Answer 34

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

Question 35

Product topology

Answer 35

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

Question 36

Open Neighbourhood( topology)

Answer 36

An open set U c X that contains x is called an open neighbourhood of x

Question 37

Neighbourhood (topology)

Answer 37

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

Question 38

Convergent (topology)

Answer 38

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

Question 39

Hausdorff space

Answer 39

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

Question 40

Connected

Answer 40

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

Question 41

Connected component

Answer 41

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

Question 42

Path connected

Answer 42

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

Question 43

Open cover

Answer 43

A collection of open sets in X is called an open cover of X if X = the union of this collection of sets

Question 44

Compact topological space

Answer 44

X is called a compact topological space if every open cover of it contains a finite subcover

Question 45

Compact

Answer 45

A subset A c X is called compact if (A,Ta) is a compact top space

Question 46

Sequentially compact

Answer 46

A metric is sequentially compact if any sequence (Xn) in X has a convergent subsequence

Question 47

e - net

Answer 47

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

Question 48

Totally bounded

Answer 48

A metric space X is called totally bounded if for any e > 0 there exists a finite e-net of X

Question 49

Normed Vector space

Answer 49

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

Question 50

Vector space

Answer 50

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

Question 51

Linear operator

Answer 51

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

Question 52

Bounded linear operator

Answer 52

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

Question 53

Continuous linear operator

Answer 53

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

Question 54

Equivalent Norms

Answer 54

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

Question 55

A banach space

Answer 55

A banach space is a normed vector space (X, II.II ) which is complete

Question 56

Convergent series

Answer 56

The formal series is called convergent to x £ X if the sequence of partial sums converges to x.

Question 57

Absolute convergence

Answer 57

The partial sums of absolute values converges to x

Question 58

Invertible

Answer 58

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.

Question 59

Homeomorphism

Answer 59

A map f is called a homeomorphism if f is bijective and both f,f-1 are continuous

Question 60

Equicontinuous

Answer 60

A is equicontinuous, that is, ∀ε > 0 ∃δ > 0 ∀f ∈ A: d(x, y) < δ =⇒ |f(x) − f(y)| < ε

Question 61

Uniformly continuous

Answer 61

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

Question 62

Maximum

Answer 62

The maximum max A to be an element c ∈ A such that x ≤ c for all x ∈ A.

Question 63

Minimum

Answer 63

The minimum min A to be an element c ∈ A such that x ≥ c for all x ∈ A

Question 64

Supremum

Answer 64

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.

Question 65

Infimum

Answer 65

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.