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

^{H}

^{x}

### 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} ^{sen}^{H} ^{O}

Exampies

### y

^{a b}

^{interval}

Ca ^{b} ^{space}

### of

^{continuos}

^{functions}

^{on}

^{cab}

### for ^{f}

^{e}

^{C} ^{laid}

^{It}

^{fell}

^{sup}

^{NE}

^{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 11}

^{p}

### 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}

^{IR}

^{2}

^{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}

^{well}

^{defined}

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

convergencep 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 ^{Jong}^{du} ^{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}

^{linmsusPJH}

^{n}

^{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}