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

By Carlo Alabiso, Ittay Weiss

This e-book is an creation to the speculation of Hilbert area, a basic device 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 area, instead of a close examine of Hilbert area concept itself, is living within the very excessive mathematical trouble of even the best actual case. inside of a regular graduate path in physics there's inadequate time to hide the idea of Hilbert areas and operators, in addition to distribution thought, with enough mathematical rigor. Compromises has to be stumbled on among complete rigor and sensible use of the tools. The ebook is predicated at the author's classes on useful research for graduate scholars in physics. it's going to equip the reader to procedure Hilbert area and, hence, rigged Hilbert house, with a more effective attitude.

With appreciate to the unique lectures, the mathematical taste in all matters has been enriched. furthermore, a short creation to topological teams has been additional as well as workouts and solved difficulties through the textual content. With those advancements, the e-book can be utilized in higher undergraduate and reduce graduate classes, either in Physics and in Mathematics.

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

Example text

For all a, b, c ∈ R with c ≥ 0: a ≤ b =⇒ a · c ≤ b · c. In general, a field K with a total order on it such that these two axioms hold is called an ordered field. For instance, Q with the usual ordering is an ordered field. For the field C, however, it can be shown without too much difficulty that no ordering of it exists that turns it into an ordered field. Finally, perhaps the most important property of the real numbers, and certainly one that sets it apart among all ordered fields, is its Dedekind completeness.

3 Let V be a linear space and B a basis for it. Then every vector x ∈ V can be expressed uniquely as a linear combination of elements from B. In other words, there is a bijective correspondence between the vectors x ∈ V and the elements of (K B )0 . Proof Since a typical element in (K B )0 is nothing but a function x : B → K for which x(b) = 0 for all but finitely many b ∈ B, we may associate with each such x the vector b∈B x(b) · b (as the summation is finite). We claim that this correspondence is the desired bijection.

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. Notice that since g is injective, the inverse image g −1 (x0 ), for any x0 ∈ X , is either empty or is of the form {y0 }, where g(y0 ) = x0 . Similarly, f −1 (y1 ), for any y1 ∈ Y , is either empty or is of the form {x1 } with f (x1 ) = y1 .

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