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

By Carlo Alabiso, Ittay Weiss

This booklet is an advent to the idea of Hilbert house, a basic instrument 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 idea of Hilbert house, instead of an in depth research of Hilbert area conception itself, is living within the very excessive mathematical trouble of even the easiest actual case. inside a standard graduate path in physics there's inadequate time to hide the idea of Hilbert areas and operators, in addition to distribution idea, with enough mathematical rigor. Compromises has to be chanced on among complete rigor and sensible use of the tools. The booklet is predicated 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, as a result, rigged Hilbert area, with a more effective attitude.

With admire to the unique lectures, the mathematical taste in all topics has been enriched. additionally, a short advent to topological teams has been further as well as workouts and solved difficulties during the textual content. With those advancements, the publication can be utilized in higher undergraduate and reduce graduate classes, either in Physics and in Mathematics.

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

Example text

Let us now denote by X ∞ all type ∞ elements in X , by X X all type X elements in X , and by X Y all type Y elements in X . Similarly, introduce the notation Y∞ , Y X , and YY . 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.

F M (JM ) ∪ {y}) is a linearly dependent set for every y ∈ S − f M (JM ). Now, since S is a spanning set we may write x! = αs · s = s∈S αs · s + s∈ f M (JM ) αs · s = x 1 + x 2 s∈S− f M (JM ) where the sum is a finite sum, so that αs = 0 for all but finitely many s, and we simply split the sum according to whether or not s ∈ f M (J M ). By our assumption, the set (I − (J M ∪ {x! })) ∪ f M (J M ) is linearly dependent if any of the vectors s ∈ S − f M (J M ) is added to it. Thus, this set is linearly dependent if any linear combination of such vectors, such as x 2 , is added to it.

12 The Cantor-Shröder-Bernstein Theorem The result we present now is a very convenient tool in establishing that two sets have the same cardinality. 1 (Cantor–Shröder–Bernstein) For all sets X and Y , if |X | ≤ |Y | and |Y | ≤ |X |, then |X | = |Y |. Proof By the condition in the assertion, there exists an injective function f : X → Y and an injective function g : Y → X . To construct a bijection h : X → Y , we consider the behaviour of elements in both X and Y with respect to the given functions f and g.

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