A Banach space in which every injective operator is surjective

TL;DR

This paper constructs an infinite-dimensional Banach space of continuous functions where every injective operator is also surjective, revealing a unique structural property of such spaces.

Contribution

It introduces a specific Banach space C(K) with the property that all injective operators are surjective, a novel example in functional analysis.

Findings

Every injective operator on C(K) is surjective.

The space C(K) has a unique operator structure.

This provides new insights into operator theory on Banach spaces.

Abstract

We construct an infinite dimensional Banach space of continuous functions C(K) such that every one-to-one operator on C(K) is onto.