stakgold chap 2 meta meta

Meta Meta Review of Chapter 2 PhL date uncertain This meta meta was originally at the front of the meta notes, but on 3.14.11 I put the meta meta notes into a separate doc (ie,. this doc) since that is done on all the other chapters. About 4.5 pages here. 2.1 Functions and Transformations (Operators). ( pp 92-96 ) 1 2.2 Linear Spaces (= Vector Spaces). ( pp 96-99 ) 1 2.3 Metric Spaces, Normed Linear Spaces, Inner Product Spaces ( pp 99-116 ) 1 2.4 Properties of Separable Hilbert Spaces (HS). ( pp 116-135 ) 2 2.5 Functionals. ( pp 135-139 ) 2 2.6 Operators. ( pp 139-146 ) 2 2.7 Operators in the Hilbert Space En(c) ( pp 146-165 ) Finite Dimension Hilbert Spaces 2 2.8 The inverse of an linear operator ( pp 165-180 ) Infinite Dimension Hilbert Spaces 3 2.9 The spectrum of an operator ( pp 180-184 ) 3 2.10 Completely Continuous Operators. ( pp 184-187 ) 3 2.11 Extremal Properties for Bounded Operators ( pp 187 - 190) 3 PL Appendix: the resolvent. 4 2.1 Functions and Transformations (Operators). ( pp 92-96 ) General notion of a mapping or "transformation" A:X→Y where X,Y are sets or spaces, but domain and range might be subsets of X and Y. If 1-to-1, can talk about inverse A-1. If spaces, can consider both finite or infinite dimensional spaces. A need not be linear. 2.2 Linear Spaces (= Vector Spaces). ( pp 96-99 ) Preview of independent vs dependent vectors, notions of dimension and of a basis. 2.3 Metric Spaces, Normed Linear Spaces, Inner Product Spaces ( pp 99-116 ) The metric space is discussed first, notion of distance and metric d(x,y) and convergence of a sequence of points in such a space which need not also be a vector space, ie, don't have to know what x+y or αx means to talk about points in a metric space. Regular and Cauchy sequence convergence ideas. Notion of space being complete, how to make it be complete. Boundary of an open set, notion of a set being sequentially compact. Notion of a set being dense in another set. Polynomials are dense in C(a,b) says the Weierstrass Approximation Theorem. Use overbar to show a completed or closed set. The normed vector space adds a norm ||x|| to a vector space, totally unrelated to metric space a priori. Length of a vector need not be related to distance between two vectors. But if d(x,y) = ||x-y|| is added as the natural metric then a normed linear space (assuming complete) becomes a Banach space, 1932. The inner product space is instead a vector space to which is added <x,y>. Usually one then uses the natural norm ||x||2 = <x,x> and from that the natural metric, and the result is (if complete) a Hilbert Space (1900-1910). The famous CSI says |<x,y>|2 ≤ <x,x><y,y>. The prototype HS is Rn, and when acted on by linear operators, you have "linear algebra" aka matrix algebra. In QM the inner product order is reversed from this book. Graphic shows all the types of spaces and how related. 2.4 Properties of Separable Hilbert Spaces (HS). ( pp 116-135 ) Separable means a countable spanning set (like tn), which is only a basis if the projected coefficients don't move as you improve a fit. In this case, they are generalized Fourier Coefficients. The projection theorem says a subspace (= linear manifold) can be treated as M M and you can write any x H as x1 + x2 etc. Orthogonal spanning set is a basis, many classical examples of such polynomials. The Riesz-Fischer Theorem (1907) makes a connection between convergence of a sequence of vectors in a HS and the convergence of a sequence of real numbers, but you have to do this with respect to a known basis in the space. The real numbers are the squares of the partial sums of the squared Fourier coefficients. 2.5 Functionals. ( pp 135-139 ) Notion of a norm ||T|| so T can be bounded over a set S. Opposite is unbounded. The definition of continuity of T in terms of sequences. Reminder of uniform convergence idea. Continuous at x=0 implies continuous in all of domain. Bounded = continuous for linear T. Riesz Representation Theorem (1907) makes isomorphism between bounded=continuous linear functionals T and elements t of H such that T[x] = <x,t>. [ δ is not bounded.] The nullspace of a functional is defined, it has a perp space. Notion of extending or restricting T relative to its domain. All functionals linear in En are bounded and continuous. The dual or reciprocal basis idea. 2.6 Operators. ( pp 139-146 ) The norm ||A|| is introduced with various theorems which are just variations of the T theorems of the previous section. Extension and restriction as well. Notion of a closable and closed operator in terms of sequences. Close the domain as well. Theorems on closed operators. Differentiation is an unbounded but closed operator. Domain is absolutely continuous functions. 2.7 Operators in the Hilbert Space En(c) ( pp 146-165 ) Finite Dimension Hilbert Spaces This long section gets at basic operator ideas but within the limited confines of En . Later we come back and redo everything more generally to handle ∞ dim Hilbert Spaces. Operator here is a square matrix, either regular or singular based on det(A). Definition of the adjoint operator written as A*. Self adjoint means A = A* (same as complex symmetric or Hermitian). The Alternative Theorem says A has an inverse so Ax=f has a solution, or A has a nullspace and RA = (NA*) . So the nullspaces eats into the dimensionality of the range. Definition of rank (dim RA) and nullity (dim NA). Rule is that rank + nullity = n. Nullspace of B = A-λI is space of eigenvectors of A and eigenvalues come from detB = 0 which is the secular equation. Notions of algebraic ki and geometric mi multiplicity of an eigenvalue, and mi ≤ ki. Every operator has at least one eigenvector. Every eigenvalue has at least one eigenvector which says mi ≥ 1. A full rank matrix A. If there are N eigenvectors all with different λi, they form a basis. A symmetric operator is an important special case: eigenvalues are real, eigenvectors of different λi are orthogonal and form a basis. No missing eigenvectors, ki = mi in each eigenmanifold. Can write H = M1 M2 M3 ..... Mk. Spectral theorem for symmetric operator lays it all out. The spectral resolution of A = Σi λi Pi. Review notions of norms I call ||A|| and |||A|||. For a symmetric operator they are equal and are equal to |λmax|. 2.8 The inverse of an linear operator ( pp 165-180 ) Infinite Dimension Hilbert Spaces This section generalizes the notions of the previous section to arbitrary (∞ dim) Hilbert Spaces. This entire chapter is only about linear operators, note from title. The title seems a bit of a misnomer, this subsection really is about redoing all our previous En theory for ∞ dim HS, but of course the inverse A-1 is always of major interest if we want to solve Ax = f. Inverse exists if there is no nullspace (meaning only the trivial nullspace). More detailed definition of A being regular: empty nullspace, range is entire HS, A-1 must be bounded = continuous. Essentially regular if you can extend the range to be all of H. The notion of closed operators is introduced, to eliminate the possibility of a finite range limit point for the image of a null sequence in the domain. I drew a second Visio picture showing how linear operators are classified in various ways. There are four distinct cases of closed operators. The adjoint operator is defined as A*, and the new version of the Alternative Theorem is stated. Adjoint might exist only for some set of admissible pairs { y, A*y=g}. Facts about adjoints are stated. We are always interested in A* and NA* because Alternative Theorem relates this to RA and hence to the solvability of Ax = f. A self-adjoint operator has more requirements than it did in the Rn world. A symmetric operator is a less restrictive thing, and this is my third Visio picture. Theorems about self-adjoint and symmetric operators. Then a huge EXAMPLES section, including the mother of all examples A = d/dt with various BC's. 2.9 The spectrum of an operator ( pp 180-184 ) We review the classification of closed operators: one regular and three singular cases. We apply this to the operator B = (A - λI) and are then able to classify the possible values of λ into four bins. If B is regular, the bin is called the resolvent set. The other three bins are singular bins their union is the spectrum, that is, the range of λ values for which B is singular. Those bins are point spectrum (for eigenvalues), the continuous spectrum for when B-1 is unbounded, and the residual spectrum where RB < H so you have some (RB) there whose dimension is called the deficiency of λ, a strange name. Theorems on spectra are given. If A is symmetric, there is no residual spectrum. The spectrum of a self-adjoint operator lies entirely on the real axis (and there is no residual spectrum). EXAMPLES are given. 2.10 Completely Continuous Operators. ( pp 184-187 ) Review of the notion of compactness, and I wrote a separate document on this. We know that compact bounded, but the reverse is not true, and the example is given of the φn orthonormal sequence. The idea is that whereas a bounded = continuous operator always maps bounded sets into bounded sets, a completely continuous operator always maps bounded sets into compact sets, and some web sites call these compact operators. Interestingly, the identity is not completely continuous. Various theorems are given . 2.11 Extremal Properties for Bounded Operators ( pp 187 - 190) This section contains 6 theorems which are mostly pretty simple. Here are some of them: Theorem 1: If A is bounded and DA = H , then || A || = || A* || . Theorem 2: If A is bounded and DA = H , then |||A||| ||A||, for all x 0. Theorem 3: If is in any of the three kinds of spectra for A, then | | || A ||. Theorem 4: If A is symmetric, |||A||| = ||A||. Theorem 6: If A is symmetric and completely continuous, then the largest eigenvalue is || A ||. PL Appendix: the resolvent. So in our notation, we usually write B = (A - λI). We might then consider an equation of the form Bx = f Ax - λx = f f = -λx + Ax // integral equation form Lu - λu = f // differential equation form Our formal solution then is x = B-1 f = (A - λI)-1f and so B-1 is called the resolvent. I suppose we have to regard this inverse operator as a power series of some sort in A, and that is probably the famous series approach one takes to Fredholm approximation. For the differential equation above we use the more familiar L in place of A. _________________________________________________________________________________