A complete locally convex space of countable dimension admitting an operator with no invariant subspaces

TL;DR

This paper constructs a complete locally convex space of countable dimension with a continuous linear operator that has no non-trivial closed invariant subspaces, challenging assumptions about invariant subspace existence.

Contribution

It introduces a novel example of a complete locally convex space with an operator lacking invariant subspaces, expanding understanding of operator theory in topological vector spaces.

Findings

Existence of an operator with no invariant subspaces on a countably dimensional space

Construction of a complete locally convex space with this property

Implications for invariant subspace problem in topological vector spaces

Abstract

We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.