Abstract
The basic concepts of finite-dimensional vector spaces introduced in Chap. 2 can readily be generalized to infinite dimensions. The definition of a vector space and concepts of linear combination, linear independence, subspace, span, and so forth all carry over to infinite dimensions. However, one thing is crucially different in the new situation, and this difference makes the study of infinite-dimensional vector spaces both richer and more nontrivial: In a finite-dimensional vector space we dealt with finite sums; in infinite dimensions we encounter infinite sums. Thus, we have to investigate the convergence of such sums.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
It is possible to introduce the idea of closeness abstractly, without resort to the notion of distance, as is done in topology. However, distance, as applied in vector spaces, is as abstract as we want to get.
- 2.
Recall that one can always define an inner product on a finite-dimensional vector space. So, the existence of orthonormal bases is guaranteed.
- 3.
We can consider |f n 〉 as an “approximation” to |f〉, because both share the same components along the same set of orthonormal vectors. The sequence of orthonormal vectors acts very much as a basis. However, to be a basis, an extra condition must be met. We shall discuss this condition shortly.
- 4.
For an elementary discussion of the Dirac delta function with many examples of its application, see [Hass 08].
- 5.
Do not confuse this with an n-dimensional vector. In fact, the dimension is n-fold infinite: each x i counts one infinite set of numbers!
- 6.
See [Zeid 95, pp. 27, 156–160] for a formal definition of continuity for linear functionals.
References
Friedman, A.: Foundations of Modern Analysis. Dover, New York (1982)
Hassani, S.: Mathematical Methods for Students of Physics and Related Fields, 2nd edn. Springer, Berlin (2008)
Reed, M., Simon, B.: Functional Analysis (4 volumes). Academic Press, New York (1980)
Simmons, G.: Introduction to Topology and Modern Analysis. Krieger, Melbourne (1983)
Zeidler, E.: Applied Functional Analysis. Springer, Berlin (1995)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hassani, S. (2013). Hilbert Spaces. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-01195-0_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01194-3
Online ISBN: 978-3-319-01195-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)