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1. Introduction Not written as of yet. Topics to mention. (1) A better and more general integral. (a) Convergence Theorems (b) Integration over diverse collection of sets. (See probability theory.) (c) Integration rela
Chapter II Elements of Functional Analysis
Lecture 8
1. Hahn-Banach Theorems
The result we are going to discuss is one of the most fundamental theorems in the whole eld of Functional Analysis. Its statement is simple but quite technical. Definitions. Let
Lecture 12
3. Banach spaces
Definition. Let K be one of the elds R or C. A Banach space over K is a normed K-vector space (X, . ), which is complete with respect to the metric d(x, y ) = x y , x, y X. Example 3.1. The eld K, equipped with the absolute val
Lecture 13
4. The weak dual topology
In this section we examine the topological duals of normed vector spaces. Besides the norm topology, there is another natural topology which is constructed as follows. Definition. Let X be a normed vector space over K(
Lectures 14-15
5. Banach spaces of continuous functions
In this section we discuss a examples of Banach spaces coming from topology. Notation. Let K be one of the elds R or C, and let be a topological space. We dene K Cb () = cfw_f : K : f bounded and con
Lectures 16-17
6. Hilbert spaces
In this section we examine a special type of Banach spaces. Definition. Let K be one of the elds R or C, and let X be a vector space over K. An inner product on X is a map XX (, ) K, with the following properties: 0, X; if
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4. The Riemann Integral In this short chapter, the Riemann integral for Banach space valued functions is dened and developed. Our exposition will be brief, since the Lebesgue integral and the Bochner Lebesgue integral will subsume the
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5. Ordinary Differential Equations in a Banach Space Let X be a Banach space, U o X, J = (a, b) 3 0 and Z C (J U, X ) Z is to be interpreted as a time dependent vector-eld on U X. In this section we will consider the
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22. Banach Spaces III: Calculus In this section, X and Y will be Banach space and U will be an open subset of X. Notation 22.1 ( , O, and o notation). Let 0 U o X, and f : U Y be a function. We will write: (1) f (x)
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11. Approximation Theorems and Convolutions Let (X, M, ) be a measure space, A M an algebra. Notation 11.1. Let Sf (A, ) denote those simple functions : X C such that 1 (cfw_) A for all C and ( 6= 0) < . P For Sf (A,
Lecture 7
7. Baire theorem(s)
In this section we discuss some topological phenomenon that occurs in certain topological spaces. This deals with interiors of closed sets. Exercise 1. Let X be a topological space, and let A and B be closed sets with the pro
Lecture 6
6. Metric spaces
In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map d : X X [0, ) with the following properties: (i) If x, y X are points with d(x, y ) = 0, then x = y ; (ii) d(x, y
1
Assume the set cfw_1 , 2 , has linearly independent vectors, then
For subspace cfw_1 , 2 , , it has vectors as basis, and has dimension of
For vector space , it is an dimensional space, which means it has an basis
with linearly independent vectors.
If
2
Since is a vector space with dimension 3, we write = cfw_1 , 2 , ,
which are linearly independent.
We know 1 , and dim(1 ) = 1, which means there are 1 linearly
independent vectors in 1 , which also contains in space
cfw_1 , 2 , , , , . .
= cfw_1 , 2
def most_popular(old_dic, start, end):
new_dic = cfw_
for cha in old_dic:
new_dic[cha] = 0
for item in old_dic[cha]:
if item[2]>=start and item[2]<=end:
new_dic[cha] = new_dic[cha] + item[4] +item[5]
compare_dic = cfw_
for cha in new_dic:
if new_dic[cha]
def remove_same(lst):
"(list) -> list
Return a list from lst with the same word remove from lst, except for the
first appearance of the word.
> remove_same(['Angela','Angela'])
['Angela']
> remove_same(['Andy', 'Paul', 'Lin'])
['Andy', 'Paul', 'Lin']
> re
import unittest
import tweets
class TestExtractHashtags(unittest.TestCase):
def test_no_hashtags(self):
" Test extract_hashtags with a tweet with no hashtags. "
actual_hashtags = tweets.extract_hashtags('this is a tweet!')
expected_hashtags = []
self.asse
import unittest
import tweets
class TestCommonWords(unittest.TestCase):
def test_none_removed(self):
" Test common_words with N so that no words are removed. "
words_to_counts = cfw_'cat': 1
expected_result = cfw_'cat': 1
tweets.common_words(words_to_coun
def is_palindrome (str1):
'(str) -> bool
Precondition: word only contains lowercase alphabetic letters.
Return True if and only if str1 is a palindrome.
>is_palindrome ('qwewq')
True
>is_palindrome ('qwewqd')
False
'
return str1 = str1[:-1]
def is_palindr
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12. Hilbert Spaces 12.1. Hilbert Spaces Basics. Denition 12.1. Let H be a complex vector space. An inner product on H is a function, h, i : H H C, such that (1) hax + by, z i = ahx, z i + bhy, z i i.e. x hx, z i is linear. (2) hx, y
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29. Unbounded operators and quadratic forms 29.1. Unbounded operator basics. Denition 29.1. If X and Y are Banach spaces and D is a subspace of X , then a linear transformation T from D into Y is called a linear transf
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6. Elliptic Ordinary Differential Operators Let o Rn be a bounded connected open region. A function u C 2 () is said to satisfy Laplaces equation if More generally if f C () is given we say u solves the Poisson equation if In order to
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7. Test Functions and Partitions of Unity 7.1. Convolution and Youngs Inequalities. Letting x denote the delta function at x, we wish to dene a product () on functions on Rn such that x y = x+y . Now formally any functi
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8. Surfaces, Surface Integrals and Integration by Parts Denition 8.1. A subset M Rn is a n 1 dimensional C k -Hypersurface if for all x0 M there exists > 0 an open set 0 D Rn and a C k -dieomorphism : D B (x0 , ) such that (D cfw_xn =
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9. Poisson and Laplaces Equation For the majority of this section we will assume Rn is a compact manifold with C 2 boundary. Let us record a few consequences of the divergence theorem R in Proposition 8.28 in this context. If u, v C
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13. Abstract Wave Equation In the next section we consider (13.1) Before working with this explicit equation we will work out an abstract Hilbert space theory rst. Theorem 13.1 (Existence). Suppose A : H H is a self-ad