Download A Primer on Hilbert Space Theory: Linear Spaces, Topological by Carlo Alabiso, Ittay Weiss PDF

By Carlo Alabiso, Ittay Weiss

This publication is an creation to the speculation of Hilbert house, a primary software for non-relativistic quantum mechanics. Linear, topological, metric, and normed areas are all addressed intimately, in a rigorous yet reader-friendly model. the reason for an creation to the speculation of Hilbert area, instead of an in depth learn of Hilbert house concept itself, is living within the very excessive mathematical hassle of even the best actual case. inside a normal graduate path in physics there's inadequate time to hide the speculation of Hilbert areas and operators, in addition to distribution thought, with enough mathematical rigor. Compromises has to be chanced on among complete rigor and sensible use of the tools. The publication relies at the author's classes on practical research for graduate scholars in physics. it's going to equip the reader to technique Hilbert area and, to that end, rigged Hilbert house, with a simpler attitude.

With appreciate to the unique lectures, the mathematical taste in all topics has been enriched. additionally, a quick creation to topological teams has been additional as well as workouts and solved difficulties in the course of the textual content. With those advancements, the e-book can be utilized in higher undergraduate and decrease graduate classes, either in Physics and in Mathematics.

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Extra resources for A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups

Example text

The claim now follows since the latter is the dimension of V . Now, if V is finite dimensional, say of dimension n, and A = {x1 , . . , xn } is a set of n linearly independent vectors, then to show A is a basis we just need to prove that it is a spanning set. But if it were not, and y ∈ V is any vector not in its span, then the set {x1 , . . , xn , y} is linearly independent and contains n + 1 vectors. But then, by the first part of the proposition, it would follow that n + 1 ≤ n, an absurdity.

Thus X = X∞ ∪ X X ∪ XY and the union is pairwise disjoint. Similarly we obtain the pairwise disjoint union Y = Y∞ ∪ Y X ∪ YY . 14 1 Introduction and Preliminaries The function given by ⎧ ⎪ if x ∈ X X ⎨ f (x) h(x) = g −1 (x) if x ∈ X Y ⎪ ⎩ f (x) if x ∈ X ∞ is now easily seen to be a bijection, for instance by constructing an inverse function for it. 2 A subset of a countable set is countable. Proof Suppose a set X is countable and S ⊆ X is a subset. If S = ∅, then it is countable, so we may assume S = ∅ and let us fix some s0 ∈ S.

The familiar definitions (x + y)(t) = x(t) + y(t), (αx)(t) = α · x(t), when applied to continuous functions x, y : I → R, are well-known to give continuous functions again, and it is easy to see that when these operations are taken as addition and scalar multiplication, the set C(I, R) is a linear space over R. One may also consider the set C(I, C) of all continuous complex-valued functions x : I → C to similarly obtain a linear space over C. One may also consider, for each k ≥ 1, the set C k (I, R) of all functions x : I → R with a continuous k-th derivative, which is similarly a linear space over R.

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