
Published online:
20 September 2018
Published in print:
27 February 2018
Online ISBN:
9781400889921
Print ISBN:
9780691178561
Contents
-
-
-
-
-
-
1. The Definition of Hilbert Space 1. The Definition of Hilbert Space
-
2. The Geometry of Hilbert Space 2. The Geometry of Hilbert Space
-
3. Digression on the Conditions A-E 3. Digression on the Conditions A-E
-
4. Closed Linear Manifolds 4. Closed Linear Manifolds
-
5. Operators in Hilbert Space 5. Operators in Hilbert Space
-
6. The Eigenvalue Problem 6. The Eigenvalue Problem
-
7. Continuation 7. Continuation
-
8. Initial Considerations Concerning the Eigenvalue Problem 8. Initial Considerations Concerning the Eigenvalue Problem
-
9. Digression on the Existence and Uniqueness of the Solutions of the Eigenvalue Problem 9. Digression on the Existence and Uniqueness of the Solutions of the Eigenvalue Problem
-
10. Commutative Operators 10. Commutative Operators
-
11. The Trace 11. The Trace
-
-
-
-
-
-
-
Cite
von Neumann, John, Robert T. Beyer, and Nicholas A. Wheeler, 'Abstract Hilbert Space', in John von Neumann, and Robert T. Beyer (eds), Mathematical Foundations of Quantum Mechanics: New Edition (Princeton, NJ , 2018; online edn, Princeton Scholarship Online, 20 Sept. 2018), https://doi.org/10.23943/princeton/9780691178561.003.0003, accessed 18 Apr. 2025.
Abstract
This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).
Keywords:
Hilbert space, abstract Hilbert space, mathematical basis, geometry, closed linear manifolds, eigenvalue problem, commutative operators
Translator:
You do not currently have access to this chapter.
Sign in
Personal account
- Sign in with email/username & password
- Get email alerts
- Save searches
- Purchase content
- Activate your purchase/trial code
- Add your ORCID iD
Purchase
Our books are available by subscription or purchase to libraries and institutions.
Purchasing informationMetrics
View Metrics
Metrics
Total Views
125
85
Pageviews
40
PDF Downloads
Since 12/1/2022
Month: | Total Views: |
---|---|
December 2022 | 2 |
January 2023 | 4 |
February 2023 | 1 |
March 2023 | 2 |
April 2023 | 3 |
June 2023 | 13 |
July 2023 | 4 |
September 2023 | 1 |
October 2023 | 2 |
November 2023 | 4 |
December 2023 | 3 |
January 2024 | 5 |
February 2024 | 13 |
March 2024 | 2 |
April 2024 | 2 |
May 2024 | 5 |
June 2024 | 10 |
July 2024 | 7 |
August 2024 | 10 |
September 2024 | 1 |
October 2024 | 5 |
November 2024 | 2 |
December 2024 | 4 |
January 2025 | 2 |
February 2025 | 5 |
March 2025 | 12 |
April 2025 | 1 |
Citations
Altmetrics
More from Oxford Academic
Get help with access
Institutional access
Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:
IP based access
Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.
Sign in through your institution
Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institution’s website and Oxford Academic.
If your institution is not listed or you cannot sign in to your institution’s website, please contact your librarian or administrator.
Sign in with a library card
Enter your library card number to sign in. If you cannot sign in, please contact your librarian.
Society Members
Society member access to a journal is achieved in one of the following ways:
Sign in through society site
Many societies offer single sign-on between the society website and Oxford Academic. If you see ‘Sign in through society site’ in the sign in pane within a journal:
If you do not have a society account or have forgotten your username or password, please contact your society.
Sign in using a personal account
Some societies use Oxford Academic personal accounts to provide access to their members. See below.
Personal account
A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.
Some societies use Oxford Academic personal accounts to provide access to their members.
Viewing your signed in accounts
Click the account icon in the top right to:
Signed in but can't access content
Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.
Institutional account management
For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.