Space upon which addition and scalar multiplication are defined such that:

for every x,y,z in X: (x+y) + z = x + (y+z)

there exists a 0_x  s.t. for every x there exists -x s.t. x+ (-x) = 0

for every k in R or C, k(x+y) = kx+ky 

|| . ||

i) || x || ≥ 0 , || x || = 0 <=> x=0

ii) ||kx|| = |k| ||x||

iii) ||x-y|| ≤ ||x-z|| + ||z-y|| 

defined only on linear spaces

a. d: X x X -> R

i) d(x,y) ≥ 0 d(x,y)=0 <=> x=y

ii) d(x,y) = d(y,x)

iii) d(x,y)≤d(x,z) + d(z,y)

b. collection of points on which a metric is defined

a. Continuous Transformation

b. sequential continuity

a. T is continuous at x₀ if

for every ε>0, there exists a δ>0 such that:

||x-x₀|| ≤ δ => ||T(x)-T(x₀)|| ≤ ε

b. T is sequentially continuous at x₀ if

x->x₀ => T(x)->T(x₀)

seq. cont. <=> cont. 

a) inner-product defined <.,.> : H x H -> C

i)<x,x> ≥ 0, <x,x>=0 <=> x = 0

ii)<x,y> = conj(<y,x>)

iii)<x,y> ≤ <x,z> + <z,y>

iv)<kx,y> = k<x,y>

b) complete ( all Cauchy sequences converge in H)

|<x,y>|² ≤ ||x||²||y||²

{x_n} Cauchy if

for every e>0, there exists N st

for every n,m≥N

||x_n-x_m||<e

Dense subset

a. English

b. math

a. subset in which all elements of another subspace can be arbirtarily closely approximated 

b. D C X is a dense subset of a normed linear space X if for all x in X, e>0:

there is a y in D st

||x-y||≤e

a. X is separable iff it contains a complete orthonormal sequence

b. X is separable iff it contains a dense countable subset

Ortho => perpendicular

 all elements of the sequence are perpendicular to   each other (<x,y> = 0)

normal => ||.|| = 1

sequence => bijection between set and N

let the norm on C([a,b]) be

||f|| = sup (t in [a,b]) of |f(t)|

the set of polynomials:

sum(1,n){a_itⁱ}, ai in R, n in Z

is dense in C([a,b])

(i.e. polynomials without constant terms can be used to closely approximate continuous functions)

sacrifice e/2 for removal of later terms (L2 f'ns go to zero)

sacrfice e/2 to approximate f with a polynomial (Wieirstrass' approx. thm)

=> less than 'e' error

a) λ(A) = int [ 1_{{x in A}}λ(dx) ] with indicator f'n

b) define simple f'n:

fn(x) = sum(1,n){ak 1_{{x in Ak}}}

  int [ fn(x)λ(dx) ] = sum(1,n){ak.λ(Ak)}

c) for every f, there is a seq. of simp. f'ns st

lim fn(x) = f(x) for every x

int [ f(x)λ(dx) ] = lim int [ fn(x)λ(dx) ] 

(indicator f'ns are dense in L2([a,b];R)

0≤fn(x)≤f(n+1)(x) and lim fn(x) = f(x)

=>

lim int [ fn(x)λ(dx) ] =int [ lim fn(x)λ(dx) ] 

= int [ f(x)λ(dx) ] 

lim fn(x) = f(x) and there is g(x) st

|f(x)|≤g(x) for every x 

and

 int [ g(x)λ(dx) ] < inf

=> lim int [ fn(x)λ(dx) ] =int [ f(x)λ(dx) ] 

ψ 0,0 = 1 for t in [0,1]

φ n,k = {2^n/2 for t in [k2^-n,(k+1/2)2^-n]

{-2^n/2 for t in [(k+1)2^-n,(k+1)2^-n]

{0 o.w.

all linear functionals on X, f such that f is bounded (and hence continuous)

for x in X*:

there is an n in lq st

f(x) = sum(1,n){xi.ni} = <x,n>

for every x in X = lp

where 1/p + 1/q = 1

Hölder's inequality

x in lp, y in lq, 1/p + 1/q = 1

=>sum (0,inf) { xi yi} = <x,y> ≤||x||_p||y||_q

hence:

if n in lq

=> sup for ||x||=1 of (<x,n>/||x||_p) ≤ ||n||_q

if xn->x in n.l. X

lim ||xn - x|| = 0 in a 

lq 

st 1/p + 1/q = 1

and p,q < inf

smooth, cmpt support

no metric defined

Schwarz Space

S = {v in T(R;R) st

sup for t in R (|t^a(d^b/dt^b)v|)<inf)}

smooth

metric defined

xn -> x in S if

lim p_a,b(xn-x)=0 for every a,b in Z

where p_a,b (v) = sup for t in R (|t^a(d^b/dt^b)v|)<inf)

there exists an f such that:

ƒ = int(-inf, inf){f(t)Φ(t)} < inf for all Φ in S

if f(Φ) = g(Φ) for all Φ in S

When does a seq. of dist'ns fn

converge to a dist'n f in S*?

if {Φn} is a seq. of dist'ns:

int(-inf,inf){Φn(t)dt} = ?

false: 

Φn≥0 for all t 

for f in C([a,b]), Φn a seq. of dist'ns

what can we say about (Φn*f)(t)?

Time index, T

if:

T is countable => DT system

else: CT system

if: input/output T

shift in input (u(t-a))

results in:

 a shift in output (y(t-a))

i.e.

if (u(t),y(t)) is in R (relation)

=> 

(u(t-a),y(t-a)) is in R

i) U, Y linear spaces

ii) Relation R is a linear subspace of UxY

s.t. ((u1,y1), (u2,y2)) is in R

=> (a*u1+b*u2, a*y1+b*y2) for every a,b in Reals

DT: x(n) = sum(Z){xm.∂(n-m)}

for linear system:

y(n) = sum(Z){h(n,m)u(m)}

if u = ∂(n-n0)

=>y(n) = h(n,n0) 

output @ t = n when input is DT ∂ @ t = n0 

h(n,m)

it can be described as:

y(n) = sum(Z){h(n,m)u(m)}

if lim T(xn) = T(x)

then:

(T(x))(n) = ?

sum(-inf,inf){x(m)h(n,m)}

h(.,.): ZxZ ->C

h(n,n0+T) = h(n-T,n0)

and the system is a convolution system

time invariance

and hence

system is a convolution system

u in l(inf) -> y in l(inf)

or 

u in L(inf) -> y in L(inf)

i.e. 

BIBO stable <=> ||h||₁<inf

turns x1,x2,...,xn into o.n. seq. e1,e2,...,en

e1 = x1/||x1||

e2 = (x2-<x2,e1>e1)/||(x2-<x2,e1>e1)||

...

en = (xn-sum(i=1,n){<xn,ei>ei)})/

||(xn-sum(i=1,n){<xn,ei>ei)})||

sum(1,inf){|rk|²}<inf 

iff ...