321 200809
Chapter 4: Hilbert Spaces
4.1 Definition. An inner product space (also known as a pre-Hilbert space) is a
vector space V over K (= R or C) together with a map
h, i : V V K
satisfying (for x, y, z V and K):
(i) hx + y, zi = hx, zi + hy, zi
(ii)
Math 209C Homework 1
Edward Burkard
April 8, 2010
6.
Lp Spaces
6.1. Basic Theory of Lp Spaces.
Exercise 1. When does equality hold in Minkowskis inequality? (The answer is different for p = 1 and for 1 < p <
. What about p = ?)
Proof. We will divide this
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8. INTRODUCTION TO BANACH SPACES
3. The Lp Space
In this section we consider a space Lp (E) which resembles `p on many aspects. After general
concepts of measure and integral were introduced, we will see that these two spaces can be viewed
as special
AN INTRODUCTION TO DIFFERENTIAL GEOMETRY
EUGENE LERMAN
Contents
1. Introduction: why manifolds?
2. Smooth manifolds
2.1. Digression: smooth maps from open subsets of Rn to Rm
2.2. Definitions and examples of manifolds
2.3. Maps of manifolds
2.4. Partition
FUNCTIONAL ANALYSIS SOLUTIONS
These are solutions to problems found in Dr. Gustav Holzegels lecture notes on
Functional Analysis. The questions correspond to the lecture notes as of March
2015, i.e., the most recent version of the notes in the spring 2015
Math 118B Solutions
Charles Martin
March 6, 2012
Homework Problems
1. Let (Xi , di ), 1 i n, be finitely many metric spaces. Construct a metric on the product space X =
X1 Xn .
Proof. Denote points in X as x = (x1 , x2 , . . . , xn ). Given x, y X define
Functional Analysis Oral Exam study notes
Notes transcribed by Mihai Nica
Abstract. These are some study notes that I made while studying for my
oral exams on the topic of Functional Analysis. I took these notes from parts
1
of the textbooks A Course in F