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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
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24. Hlder Spaces Notation 24.1. Let be an open subset of Rd , BC () and BC () be the bounded continuous functions on and respectively. By identifying f BC () with f | BC (), we will consider BC () as a subset of BC (). For u BC () an
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
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
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
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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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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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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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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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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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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
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
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
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(
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
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 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
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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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12. Heat Equation The heat equation for a function u : R+ Rn C is the partial dierential equation 1 (12.1) t u = 0 with u(0, x) = f (x), 2
and hence that u(t, ) = et| /2 f ( ). Inverting the Fourier transform then sho
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20. Fourier Transform The underlying space in this section is Rn with Lebesgue measure. The Fourier inversion formula is going to state that n Z Z 1 (20.1) f (x) = deix dyf (y )eiy . 2 Rn Rn If we let = 2, this may b
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1. 2nd order differential operators Notations 1.1. Let be a precompact open subset of Rd , Aij = Aji , Ai , A0 BC () for i, j = 1, . . . , d, p(x, ) := and
d X
Aij i j +
i,j =1 d X
d X i=1
Ai i + A0
L = p(x, ) = We also let L =
i,j =1 d X
Aij i j +
d X
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21. Constant Coefficient partial differential equations P Suppose that p( ) = |k a with a C and 1 (21.1) L = p(Dx ) := |N a Dx = |N a . x i Then for f S c Lf ( ) = p( )f ( ),
that is to say the Fourier transform take
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35. Compact and Fredholm Operators and the Spectral Theorem In this section H and B will be Hilbert spaces. Typically H and B will be separable, but we will not assume this until it is needed later. 35.1. Compact Ope
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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
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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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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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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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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