Abstract
Let (H, ∥ . ∥) be a normed space. A sequence (f n) in H is said to be convergent if there exists an f ∈ H such that ∥f n − f∥→0 as n→∞. There exists at most one f ∈ H with ∥f n − f∥→0; since from ∥ f n − f∥→0 and ∥ f n − g∥→0 it follows that ∥f - g∥ ⩽ ∥ f − f n∥ + ∥ f n − g∥→0, thus f = g. We say that the sequence (f n) tends to f and call f the limit of the sequence (f n). In symbols we write f = limn→∞f n or f n→f as n→∞. If no confusion is possible, we shall occasionally abbreviate these by writing f = lim f n, or f n→f
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1980 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Weidmann, J. (1980). Hilbert spaces. In: Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6027-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-6027-1_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6029-5
Online ISBN: 978-1-4612-6027-1
eBook Packages: Springer Book Archive