j by of

Problem session ¹

Topics

^{to be covered}

today

Banach _spaces

^example

^1st

Absolute

convergence

of

^{series in}

Banach spaces

Lebesgue

Dominated

convergence theorem and applications

Differential

^{under Integra}

Picard Lindelof

^theorem

⁸ⁿ

Uniqueness Existence ^{to d VP}

of

^ODE

Banach fixed ^point

^theorem

I

Banach ^spaces

X

^vector

^space

^{over 112 6}

norm 11.11

scalars

is

^{a Banach}

space g

^{X H il is}

complete

under the metric

induced by

^{H H}

¹

every

^Cauchy

e x

y

^Hx

y

^{H X}

y X

_sequence ^{can mix}

converges lie X

sen is Cauchy

j ^F

^{N s t H m n z N}

Il ^{Nm senH O}

Exampies

y

^{a b interval}

Ca ^{b space}

of

^continuos

^functions

^{on cab}

for ^f

^e

^{C laid}

^It

^fell

^{supNE aib}

^fan

Kalb

^{H Il is a Banach}

^space Cfn

⁰

f

^e

^{C card}

2

lt ^lepta

^{sequence spaces}

Z

^{3 32 Bi c}

IR

113

_Hp

13,112,1gal't

^t

et

³

I ^11311ps

angle inequality 113 12111ps

^{11311 11 11p}

Minkowskiinequality

f

GE.iq ^{ix f EEE.is.it}

CEIrx.ij

Holder's

inequality

713 ^{x IE IE 13.15 27,1}

10

t

where

GEIR

^{s t}

pl

¹

^Iq

^L

O

Completeness sequence in

lb Cauchy

a

In

f

i is ⁱ

audrey Cauchy

Saa 3ozcat^{Y 3am c}

It

cSo F N ^{s t f mm} ZN

Il

^{3am Bn}

Kp

^{L E}

If

^13mg

^{3njlt.LE o.ly mj}

^3njl

I

It

^{is complete}

3

Lt

^spaces

^lets

^Lebesgue

^spaces

S E ^{M be a measure} space

suppose ^I

r

algebra

^{on S}

E

^c

PCs

i S EE

M measure ^µ E RU ^to _{i A EE} ^{SIA EE}

Iii

If

^{Ai EE}

U Ai ^E E

Lt ^f

^S

RI _f ^f

_Hp^is

^measurable _gift ^tea

norm

Lt ^L type ^{f t} _Http

^{S IR2 measurable}

Are ^f

^S

^{IR f}

O ^u almost everywhere

AT g ^f g

^mane

f

^S

^IR Hflls ^{f measurable int} 0201 ^Haole

^C

a ^{e x}

essential

supremum Minkowski's inequality

f ^{Hf 1gHp}

^E

^Http

111911

Holdersinequality

Hfgll

^I

Http ^11911g

where

f ^Iq

^L

Lf ^{integrable functions}

4

S

^{D E}

Rn

open

Heuer ^t

^c

⁰¹¹²¹ f

^c

^tick

for

^{all Kcomepa}

locally

J

Toc

Cr locally

^integrable

functions

j ^{Iff Banach}

^space

K

S V

^V

finite dimensional

^{v s}

u Ya ^basis

of

^V

f JE f ^y fj ^{S R}

f

^s

^f

^{G Y}

t

Fa

^{Vz i r t}

^f

^G

^Vip

J ^{f du tjd T}

Bochnerintegral

o_0 is

finite

^dim

c

Hollz ^E Il ^Il

^{E C}

^Il Hz

^{Il Il 11}

¹¹²

are

equivalent

Jfdu

^{is welldefined}

2

Absolute

convergence

of

^series

ⁱⁿ

^Banach

X

^{Banach Space Hall}

xn sequence ni

X

Sn

^{X Xz 1 Xn tf n}

Sn Sn ^{ni series}

Sn converges in

X if

^{it converges to some}

S ^E

X

Sn

absolutely

^converges

g E ^{1 1}

theorem ¹

If

^{X is a Banach}

^space

then

every

absolutely

^{convergent series}

is

convergent

2

If ^every

absolutely convergent

series

converges ^{in some}

normed ^{space X}

then

X must

^{be a}

Banach space

Proof

D

^{X is Banach}

²¹¹

^any

is convergen

n I

can

lanqgnisomersenteeny

E ^Han

^{is convergent}

^{in 112}

^{0 Luann is Cauchy}

H

^{E o F N s t F}

m n Z N

Eme

aah EI

^{ar LE}

for

^{the same}

^N

^{mi n Sak}

III

^a

^{E 941 11.77}

m

⁹

e

E Z ^{Hak 11}

K ntl

triangle

^E

L

^E

by

Snf

^{a is a}

^Cauchy

^sequence

in

X

i 0

X

X

^{norm v s Let's can}

be ^a

Cauchy

sequence bi

X

^{D F}

N

^s't

tt nk

ZN Inna ^{Hnk I}

²

Lk

^{is a} subsequence

form

^{this sequence}

of

^can

Jk ^{Knw Nnn Y}

Nn

Y

^{z Anz Nh}

Eye

^is

^{absolute convergent}

Ely ^a1

^{c D}

21

^E

D

Eye

convergence

p subsequence

ye of

^the

^Cauchy

^sequence

sen

which

_converges D sen ^{se C}

X

we

X ^Banach Space

Lebesgue

Dominated

convergence theorem

X

^{u measure}

^space

Fn

^X

^{IR measurable} functions

^{f new}

f

^{n x}

f

^Cx

^f

^X

^IR

for

^{a e X}

suppose

^F

g

^{X R}

^integrable get ⁴¹⁴

s t

Ifn

^a

^I

^E

ga ^t

^{n a e x}

then

fne ^{11cm ft L CH}

and Liz ^{th du} ft

^Dae

Proof

Fatou's

^{Lemma Suppose}

^th

measurable functions

^w

fn

²⁰

If

^nlimatncx

fcx

^{are x}

then

Jf

^E n

^limint

a

^Ifn

Want

Hnl

^{du LA}

1 Fnl Eg ^ginn

IfnldME g ^dues

monotonicity

of

integral 0

f

^{n e}

^L

^te

fnCx f

^{Cx a e x}

Ifl

^E

g

^{a e}

^SHI

da

Efg

^{du a}

f

^c

^L

^su

Want ^{to prove} fig ^Jfnd

^u

^{If die}

we ll

prove

^nling

J ^{Ifn ft du}

^o

Ifn fl

^E

^{Itnlt Ifl} 2g

^{a e}

Consider

hn 2g ^{Ifn fl}

^Z

O f ⁿ

linminffhn ^limint Sag

Stott

measurable ^{Jongdu limsupJHnf}_dm

0

ⁿ

Fatou's

^lemma

Jeniminathndustimiant

hindu

il

129dm

^YinguPSHTI

2g

^die

lim sup

g ^{Ifn ft du}

^E

^O

n a

O E

bjmijf Jlfn ^fldnf

^linmsusPJHn

^tlduf.co

Is ^{Jtfn ft du}

^O

^BO

OE µ ^Indu Adal

E Jl ^{tn ft du}

O

limo Jfndµ ^{Sf die Slim.tn}

dy

n

Counterexample

oil

fn

^{x n}

o see

In

0 otherwise

In ⁰ Ifn ^Cx _any ^{g integrable for}

Ling ^fncx3dx

^y

9

things

forex

^{dx o}

Differential ^under

^{the integral}

sign