Summary of topics

Set Theoretic Structure

Definitions

• set
• eg: numbers, sequences, functions
• elements of set
• defining a set
• list all elements
• rule
• finite/infinite
• countable/uncountable
• equality
• subsets
• contained in a set
• proper subset/superset
• disjoint

Basic set operations

• union
• intersection
• difference/symmetric difference
• universal set
• complement of a set
• demorgan's law
• cartesian products/ordered pairs
• ordered n-tuple
• useful sets
• N - natural numbers
• Z - integers
• Q - rational
• R - real
• extended real set (includes -/+ infinity)
• C - complex
  
• (more from notes) - TODO
• bounded from above/below
• upper bound/lower bound
• maximum/minimum
• least upper bound
• greatest lower bound
• supremum
• infimum

Partition & Equivalence

• Partition of X
  
• cutting into disjoint subsets of x
• two rules (TODO)
• Relation on X
• xRy
• x is related to y under relation R
• reflexive
• xRx
  
• symmetric
• xRy -> yRx
  
• transitive
• xRy & yRz -> xRz
• equivalence
• all the above properties
• x~y
• Relation and Partition
• "equivalence relation is one way of partitioning a set"
• equivalence class
• Cx, [x]
• {y in X: y~x}
• if R is an equivalent relation of X, then the family of {Cx} forms a partition on X for all x in X
• Any partition on X defines an equivalence relation on X

Functions

• other names
• mapping, transformation, operator
• f:X->Y
  
• Real valued functions of real variable
• generalized definition of functions
• function between 2 sets
• f, f()
• function
• f(x)
• element of Y assigned to x in X
• equality of functions vs. equal representation of function
• domain of f
• D(f)
• extension of f
• restriction of f
• image of x/value of f at x
  
• pre-image of y
• range of f
• R(f)
• mapping into Y
  
• R(f) contained in Y
• mapping onto Y
• R(f) = Y
• constant function
• identity mapping on X
• one-to-one mapping on X
  
• f(x1) = f(x2) -> x1 = x2
• "has only one pre-image"
• composition of g and f
• (gf)(x) = g(f(x))
• "mapping from x to z"
• exists if D(g) is contained in R(f)
• graph of f
• all ordered pairs (x,y) such that y = f(x)
  
• subset of X x Y (cartesian product)
• set function
• function for subsets??
• TODO
  
• inverse set function
• TODO
• not the same as inverse of f
• Inverse
• g is inverse of f if (gf)(x) is identity mapping on x and (fg)(y) is identity mapping on y
• inverse of f is unique
• f is invertible IFF it's one-on-one mapping and R(f) = Y (onto)
• Inverse set function vs. Inverse function
• TODO
• Left invertible
• G = F-1e
• TODO
• Right invertible
• TODO

System Types

• Time invariance
• only effect of translation in time of an input is same translation in time of the output
• discrete case
• set closed under translation
• right-shift operator/composition of right-shift operators
• continuous case
• shift operator
• Causality
  
• output independent of future inputs
• discrete case
• continuous case
• Anti-causality
• TODO

Topological Structure

Introduction to Topology

• Topology in mathematics
• point set/general topology
• i.e. topology of metric spaces
• closeness <-> metric

Metric Spaces

• Metric space
• set and metric (X,d)
• underlying set X
• metric, real valued function of x,y in X
• conditions
• positive
• strictly positive
• symmetry
• triangle inequality
• unlimited number of metrics can be defined
• Distance between x and subset A
• d(x,A) = inf{d(x,y): y in A}
• Diameter of subset A
• diam(A) = sup{d(x,y): x,y in A}
• Bounded subset

Examples

• metrics on R^2
  
• dp(x,y) = {|x1-y1|^p + |x2-y2|^p}^1/p
• Manhattan norm
• p = 1
• Euclidean norm
• p = 2
• Infinity norm
• p = inf.
• max{|x1-y1|, |x2-y2|}
•  {d = 0 if x=y, d = 1 if x!=y}
• metrics on R^n
• same ideas above extended to n-dimension
• more, TODO

Subspaces & Product Spaces

• Subspaces
• topological structure on subset A inherited from (X,d)
• if A is subset of X, then (A,d) is a metric space
• proper subspace if A!=X
• Product spaces
• metric on Z = X x Y in terms of dx and dy
• where (X,dx) and (Y,dy) are metric spaces
• (X,dx) x (Y,dy)
• Examples
• {dx(x1,x2)^p + dy(y1,y2)^p}^1/p
• d1(u,v), d2(u,v), dinf(u,v)
• n-tuple product spaces
• same idea above extended to n-dimension

Continuous functions

• continuity on F:R->R
• continuity at x0: for every e>0, d exists such that when |x-x0| < d, then |F(x) - F(x0)| < e
• F continuous if above statement is true for all x in domain
• uniform continuity in d(x0,e) -> d(e)
• continuity on F:(X,d1) -> (Y,d2)
• same as above: when d1(x,x0) < d, then d2(y,y0) < e
• Invertibility
• there's no relation between invertibility and continuity
  

Convergent functions

• Convergence for sequence of real numbers
• x0: limit of sequence {xn} = {x1, x2, ...}
• for each e>0, N exists such that n>=N -> |xn-x0|<e
• Convergence for sequence {xn} in (X,d)
• same idea as above
• x0 exists in (X,d), for each e>0, n>=N -> d(xn,x0)<e
• then we can say that lim(n->inf) xn = x0
• Only one limit exists for a convergent sequence

Continuity and Convergence

Interchanging limits and function

• sequence {xn} with limit x0
• lim(F(xn)) = F(lim(xn)) IFF F is continuous at x0
• convergent sequences {xn} in (X,d1)
• F(lim xn) = lim F(xn) IFF F is continuous
  
• i.e. mapping is contineous IFF it preserves convergent sequences

Local Neighborhoods

Continuity and convergence are "independent" of the metric used. defined in terms of set-theoretic terminology

Definitions

• Open ball
  
• centered at x0 of radius r=(0,inf)
• Br(x0) = {x in (X,d): d(x,x0) < r}
• Closed ball
• centered at x0 of radius r=[0,inf)
• Br[x0] = {x in (X,d): d(x,x0) <= r}
• Sphere
• centered at x0 of radius r=[0,inf)
• Sr[x0] = {x in (X,d): d(x,x0) = r}
• Example X = R3 with euclidian d(x,y) and x0 = (0,0,0)
  
• boundary of ball - Sr[x0]
• interior of ball - Br(x0)
• boundary + interior - Br[x0]
• Local neighborhood of x0 in (X,d)
• subset N where N = Br(x0) or Br[x0]
• open local neighborhood
• closed local neighborhood
  
• radius r != 0
• Local neighborhood system of x
• N(x): all open and closed balls of nonzero radius centered at x
• {xn} eventually in subset A contained in X
•  N exists such that xn is in A for n>=N

Properties

• for every x in Br(x0)
• There exists an Nx that's contained in Br(x0)
• Nx = local neighborhood of x
• Not necessarily true for closed local neighborhoods
• continuity of F:(X,d1) -> (Y,d2) at x0
  
• IFF inverse image of every local neighborhood of F(x0) contains a local neighborhood of x0
• inverse image - using inverse set function
• IF inverse image of every local neighborhood of F(x0) contains a local neighborhood of x0
• convergence of {xn} in (X,d) to x0
  
• IFF {xn} is eventually in every local neighborhood of x0
• IF {xn} is eventually in each local neighborhood of x0

Open Sets

asf

Closed Sets

adf

Completeness

• examples of sequences {xn} that have a limit not in the metric space X
• {xn} is a cauchy sequence
• metric space is not complete
• has a "hole", the limit is "missing"
  
• Cauchy sequence
• {xn} in (X,d)
• N exists such that for a given e>0
• n,m>= N ---> d(xn,xm)<=e
• lim as n,m->inf of d(xn,xm) = 0
• convergent sequence in (X,d) --> cauchy sequence in (X,d)
• converse is not necessarily true
• Complete (X,d)
  
• each cauchy sequence in space is a convergent sequence in space
• if we have a complete space, then proving sequence is cauchy is enough to prove that sequence converges
• Examples

Contraction Mapping

a

Algebraic Structure

Introduction

• Concept of linear spaces/ vector spaces
• Not Euclidean geometry stuff seen in math problems
• Other algebraic spaces of interest

Linear Spaces and Subspaces

Linear Spaces

• Linear space
• set + scalar field + structure
• underlying set
• scalar field: R or C in general
• structure: addition, scalar multiplication, etc.
• (X,F,+,*)
• Properties of Linear spaces
  
• addition mapping
• Multiplication mapping
• Conditions to be met for Linear spaces
• TODO p161

Linear Subspaces

• Linear subspaces
• Y is subspace of X if
  
• subset Y of linear space X
• for x1, x2 in Y, x1+x2 and ax should be in Y
• Also, Y itself is a linear space

Linear Transformation

"special case of transformations similar to how continuous transformations played a role in metric spaces"
Note: infinite series not meaningful here (based on linearity alone)

• Transformation L:X->Y, where X, Y are linear spaces over field F
• Properties of L
• L(ax) = aL(x)
• L(x1+x2) = L(x1) + L(x2)
• "operation in x -- operation in y"
• Null space of L
• subset of X: Lx=0
• linear subspace of X
• Range space of L
• subset of Y: Lx=y
• linear subspace of Y
• Linear transformation IFF principle of superposition holds

Inverse Transformation

• Linear transformation is one-to-one IFF nullspace is trivial
• If inverse of linear transformation exists, then it's linear
  

Isomorphism

• Linear spaces X,Y over field F are isomorphic if there exists
• a one-to-one linear mapping T of X onto Y
• T: isomorphism of X onto Y
• Concept of one-to-one correspondence + preservation of structure
• "central topic for equivalence of mathematical structures"
• TODO

more TODO

• contineous mapping + linear mapping
• addition & scalar mult. are contineous
• orthogonality, hilbert spaces (generalzation of Euclidian geometry)

Banach Spaces

• Real Euclidian plane
• euclidian length
• ||x||
  
• euclidian distance between x and y
• ||x-y||
• ||.|| is a distance function or metric
• here, its a "norm" on linear space
• Norm on X
• concept of above discussed function to linear spaces in general
• real valued function defined on linear space X that satisfies:
• positivity
• triangle inequality
• homogeneity
• positive definiteness
• Normed linear space
• (X,||.||)
• X is a linear space
• ||.|| is norm defined on X
• ||x-y||
• = d(x,y), a metric defined on X
• axioms of norms satisfy axioms of metrics
• i.e. norm is a metric
• equivalent norms
• different norms that have equivalent metrics
• norm is a continuous function from X to R
• therefore, addition and scalar mult. operation in X are continuous
• Banach space
• a normed linear space that is complete

Examples
examples from metric spaces that are also normed linear spaces

• norm on Rn and (Cn)
  
• ||x||p
• ||x||inf
• banach spaces
• lp set of sequences
• usual norm
• banach space
  
• BC(T,R)
• sup-norm
• banach space
• Lp
• ||.||p TODO
• usual norm
  
• banach space
• Linf
• essential supremum
  
• V(f)
  
• Function of bounded variation
• total variation of f
• more examples
• TODO

Sequences and Series

deal with concept of infinite series

• {xn} sequence in (X,||.||)
• infinite series
• sum(n = 1 to inf){xn}
  
• {ym} sequence in (X,||.||)
• sequence of partial sums
• sum(n = 1 to m){xn}
• convergence of sequence {ym}
• ym->y as m->inf
  
• IFF ||ym - y||->0 as m->inf
• convergence of infinite series
  
• if y exists, then infinite series converges
• y = sum(n = 1 to inf){xn}
• otherwise, infinite series is divergent
• convergence of infinite series in banach spaces
• use cauchy test for convergence
• this way we don't need to know what y is
  
• converges IFF for every e>0, N exists such that (n,m>=N)
  
• sum(i = n to m){xi}<=e
• absolute convergence of infinite series
• def: series of absolute values sum(n = 1 to inf){||xn||} is convergent
• ||xn|| is an element of R
• R is complete
  
• absolute convergence IFF for every e>0, N exists such that (n,m>=N)
• sum(i = n to m){||xi||}<=e
• convergence and absolute convergence in banach spaces
• if series is absolutely convergent, then it's convergent
• Examples
• 1
• 2
•