**Mathematical Foundations of System Theory**

http://www.ee.washington.edu/class/510/2011aut/index.html

Summary of topics

### Introduction to Topology

### Metric Spaces

## 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

## Combined Structure

### Introduction

- 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 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