Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET X: PROBLEMS (3, 4)
PATRICK RYAN
Problem 1. Let V be a Banach space. Show that the dimension of V is either
nite or uncountable (that is, V does not have a countably innite basis).
Proof. Let V be a Banach space, and suppose that it has a count
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Hans Prakash
4/7/16
IB HL Biology
Mr. Ajerman
Transport Lab DialysisOsmosisDifusion
Introduction
In this lab we will observe the process of diffusion and osmosis in a model of a
membrane system. This lab links to the broader theme of how diffusion works
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Name: _
IBSL Lang and Lit  1
Short Cuts Reading Log
Title:
Summary and Significance
1. In your own words, explain what happens in this story. Who were the main
characters and events? What was the climax? How did it end?
Memorable Lines
1. What lines stan
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
The Apartheid Laws
Task:
1. Read the details about each Act in your textbook and use the information to complete the second
column with the correct titles from this list:
Population Registration Act, Bantu SelfGovernment / Authorities Acts, Group Areas A
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment One Solutions
Stephen Mackereth
Problem One.
Let f : [0, ] R be a continuous function such that f (0) = f ( ) = 0, and dene real
numbers a1 , a2 , . . . by the formula
2
sin(nx)f (x)dx.
an =
0
2
n>0 (an )
Show that the sum
converges (h
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114, Problem Set 10 (due Monday, November 25)
November 16, 2013
(1) Let X and Y be metric spaces, let f : X Y be a function, and let (f ) = cfw_(x, y ) X Y : f (x) = y
be the graph of f . Show that if (f ) is closed and Y is compact, then f is conti
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Midterm (with solutions)
October 16, 2013
(1) Let C = cfw_(x, y ) R2 : x2 + y 2 = 1 be the unit circle. Show that C has measure zero (when regarded
as a subset of the Euclidean plane R2 ).
For each positive real number r, dene a function r on mea
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment Four Solutions
Stephen Mackereth
Problem One.
Suppose that f = 0 a.e. on E . Let E be any set E E , m(E ) < , and let g be any
bounded measurable function g : E R with 0 g f (since f is nonnegative we can
take g nonnegative as well).
S
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment Two Solutions
Stephen Mackereth
Problem One.
Let S Rn be a measurable set with m(S ) < and let > 0 be a positive real number.
Show that there exists a compact subset K S such that
m( S ) m( K ) m( S ) .
Proof. If S is measurable, then
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
Math 114 Assignment 3 Solutions
Stephen Mackereth
Problem 3.
Composition of Borel measurable functions is Borel measurable
Proof. It suces to show that, if
t R,
f 1 (, t]) = cfw_x R : f (x) t is Borel,
then
for all Borel sets E R,
f 1 (E ) is Borel.
But i
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET III, PROBLEMS I, II
PATRICK RYAN
Problem 1. Let E Rn be a measurable set with (E ) < 1. Show that for each
> 0, there exists a set E 0 Rn which is a nite disjoint union of open boxes
satisfying
(E E 0 ) , (E 0 E ) < .
Proof. From Problem Set
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET V: PROBLEMS 1, 2
PATRICK RYAN
Problem 1. Let E Rn be a measurable set, and let f0 f1 f2 be an
increasing sequence of integrable functions on E for which the sequence of integrals
f
is bounded. Show that the sequence cfw_fi converges almost ev
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET VIII: PROBLEMS III, IV
PATRICK RYAN
Problem 1. Let f : V ! W be a linear map between normed vector spaces. Show
that if V is nitedimensional, then f is continuous.
Proof. Let cfw_e1 , ., en be a basis for V. Then
kf (v )kW = kf (v1 e1 + + vn
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET VI SOLUTIONS (3,4)
PATRICK RYAN
Problem 1. Let f : R2 ! R be the function given by
8
>1
if (9n 2 Z 0 ) [n x, y < n + 1]
<
f (x, y ) =
1 if (9n 2 Z 0 ) [n x < n + 1 y < n + 2]
>
:
0
otherwise.
For each x 2 R, let fx denote the function given by
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET IV: PROBLEMS III, IV
PATRICK RYAN
Problem 1. Let f : Rn ! R be a measurable function. Show that for each > 0,
there exists a continuous function g : Rn ! R such that the set cfw_x 2 Rn : f (x) 6= g (x)
has measure < .
Proof. In this problem we
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Fall 2013
PROBLEM SET VII: PROBLEMS (1, 2)
PATRICK RYAN
Problem 1. Let V0 , V1 , V2 , V3 , . be real vector spaces with norms
Given an element ~ = (vn )n
v
kkn : Vn ! R
2 n 0 Vn , let
X
k~ k =
v
kvn kn 2 R
0.
0
n0
0
[ cfw_1 .
Let V n 0 Vn be the subset consisting o
Analysis II: Measure, Integration and Banach Spaces
MATH 114

Spring 2016
Vacaciones
Hans Prakash
Mi familia y yo somos de la India. India esta en
Asia. Hace dos semana mi madre, mi padre y yo
fuimos a la India. Para vacaciones fui a la India.
Paintball, ping pong, gokarting, water ballons
fights, poker nights, video games, fa