# Properties of an infinite dimensional Banach space over the field with two elements

## TL;DR

This paper demonstrates that over the field with two elements, there exists an infinite dimensional Banach space where every bounded linear operator attains its norm, addressing a longstanding open question.

## Contribution

It proves the existence of an infinite dimensional Banach space over GF(2) in which all bounded linear operators attain their norm, a previously unresolved problem.

## Key findings

- Existence of such Banach space over GF(2)
- All bounded linear operators on this space attain their norm
- Addresses a question posed by M.I. Ostrovskii

## Abstract

A banach space X is a normed vector space, which is complete with respect to the metric induced by the norm. Given a bounded linear operator T acting on a banach space X, T is said to attain its norm if there is a unit vector z in X, such that the norm of Tz equals the norm of T. The existence of an infinite dimensional banach space X, in which each bounded linear operator acting on X attains its norm, is still undetermined. This question was posed by M.I. Ostrovskii at St. John's University. In this paper we show that if an infinite dimensional banach space is considered over GF(2), then it is possible for every bounded linear operator to attain its norm.