CHAPTER 2
2.1
DYNAMIC MODELS AND DYNAMIC RESPONSE
Revisiting Examples 2.4 and 2.5 will be helpful.
(a)
R (s) = A;
Y ( s) =
y(t ) =
(b)
KA - t / t
e
;
t
R(s) =
A
;
s
KA
t s +1
yss = 0
Y(s) =
KA
s (s + 1)
y(t) = KA (1 e t / ) ;
(c)
R(s) =
yss = KA
A
KA
; Y(
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
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
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
A NALYSIS TOOLS W ITH APPLICATIONS
1
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
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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,
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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
ANALYSIS TOOLS WITH APPLICATIONS
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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
Chapter I Topology Preliminaries
Lecture 1
1. Review of basic topology concepts
In this lecture we review some basic notions from topology, the main goal being to set up the language. Except for one result (Uryson Lemma) there will be no proofs. Definitio
Lecture 2
2. The Concept of Convergence: Ultralters and Nets
In this lecture we discuss two points of view on the notion of convergence. The rst one employs a set theoretical concept, which turns out to be technically useful. Definition. Suppose X is a xe
Lecture 3
3. Constructing topologies
In this section we discuss several methods for constructing topologies on a given set. Definition. If T and T are two topologies on the same space X , such that T T (as sets), then T is said to be stronger than T . Equ
A NALYSIS TOOLS W ITH APPLICATIONS
493
27. Sobolev Inequalities 27.1. Morreys Inequality. Notation 27.1. Let S d1 be the sphere of radius one centered at zero inside Rd . For a set S d1 , x Rd , and r (0, ), let x,r cfw_x + s : such that 0 s r. So x,r = x
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23. Sobolev Spaces Denition 23.1. For p [1, ], k N and an open subset of Rd , let
k,p Wloc () := cfw_f Lp () : f Lp () (weakly) for all | k , loc
W k,p () := cfw_f Lp () : f Lp () (weakly) for all | k , (23.1) and (23.2) kf kW k,p
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11. Introduction to the Spectral Theorem The following spectral theorem is a minor variant of the usual spectral theorem for matrices. This reformulation has the virtue of carrying over to general (unbounded) self adjo
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14. Wave Equation on Rn (Ref Courant & Hilbert Vol II, Chap VI 12.) We now consider the wave equation (14.1) According to Section 13, the solution (in the L2 sense) is given by p sin(t 4) g. (14.2) u(t, ) = (cos(t 4)f
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19. Weak and Strong Derivatives For this section, let be an open subset of Rd , p, q, r [1, ], Lp () = Lp (, B , m) and Lp () = Lp (, B , m), where m is Lebesgue measure on BRd loc loc and B is the Borel algebra on . If = Rd , we wil
PDE LECTURE NOTES, MATH 237A-B
BRUCE K. DRIVER Abstract. These are lecture notes from Math 237A-B. See C:\driverdat\Bruce\CLASSFIL \257AF94 \course.tex for notes on contraction semi-groups. Need to add examples of using the Hille Yoshida theorem in PDE. S
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3. Fully nonlinear first order PDE In this section let U o Rn be an open subset of Rn and (x, z, p) U Rn R F (x, z, p) R be a C 2 function. Actually to simplify notation let us suppose U =Rn . We are now looking for a solution u : Rn R
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4. Cauchy Kovalevskaya Theorem As a warm up we will start with the corresponding result for ordinary dierential equations. Theorem 4.1 (ODE Version of Cauchy Kovalevskaya, I.). Suppose a > 0 and f : (a, a) R is real analytic near 0 and
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5. A very short introduction to generalized functions Let U be an open subset of Rn and (5.1) denote the set of smooth functions on U with compact support in U. Denition 5.1. A sequence cfw_k D(U ) converges to D(U ), i there is a k=1
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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 =