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1 - Vector Spaces and Bases
from Part I - Preliminaries
James C. Robinson, University of Warwick
Book:

An Introduction to Functional Analysis

Published online:

21 March 2020

Print publication:

12 March 2020, pp 3-16
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Summary

We recall some of the basic theory of linear algebra, beginning with the formal definition of a vector space. We then discuss linear maps between vector spaces and end by proving that every vector space has a basis using Zorn’s Lemma.

DISTRIBUTION OF GAPS BETWEEN THE INVERSES $\mathrm{mod} q$
C. Cobeli, M. Vâjâitu, A. Zaharescu
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 1 / February 2003

Published online by Cambridge University Press:

27 January 2003, pp. 185-203
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Let $q$ be a positive integer, let $\mathcal{I}=\mathcal{I}(q)$ and $\mathcal{J}=\mathcal{J}(q)$ be subintervals of integers in $[1,q]$ and let $\mathcal{M}$ be the set of elements of $\mathcal{I}$ that are invertible modulo $q$ and whose inverses lie in $\mathcal{J}$. We show that when $q$ approaches infinity through a sequence of values such that $\varphi(q)/q\rightarrow0$, the $r$-spacing distribution between consecutive elements of $\mathcal{M}$ becomes exponential.
AMS 2000 Mathematics subject classification: Primary 11K06; 11B05; 11N69

Inverses of extremal processes
Sidney I. Resnick
Journal:

Advances in Applied Probability / Volume 6 / Issue 2 / June 1974

Published online by Cambridge University Press:

01 July 2016, pp. 392-406

Print publication:

June 1974
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The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.