Nuclearity and Banach spaces

Banach Spaces and Hilbert Spaces

A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exists non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a ra...

Banach Spaces

(A revised and expanded version of these notes are now published by Springer.) 1 Banach Spaces Definition A normed vector space X is a vector space over R or C with a function called the norm. 1. The set of real numbers with the norm taken to be the absolute value.

Banach Spaces

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/05 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-example: non-existence of norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theore...

Girths and Flat Banach Spaces

J. J. Schâffer [3] introduced an interesting parameter for normed linear spaces. It is termed girth and is the infimum of the lengths of all centrally symmetric simple closed rectifiable curves which lie in the boundary of the unit ball. More precisely, let X be a Banach space with norm denoted by || -|| and with dim X^2. K curve in X will be a rectifiable geometric curve defined by Busemann [l...

Banach sequence spaces