Each normalized state is a member of an orthonormal basis: A simple proof

TL;DR

This paper provides an accessible proof demonstrating that in finite-dimensional Hilbert spaces, every normalized state can be included in an orthonormal basis, simplifying understanding for physics students.

Contribution

It offers a more comprehensible proof of a fundamental linear algebra fact relevant to quantum mechanics.

Findings

Every normalized state is part of an orthonormal basis.

The proof is tailored to be more understandable for physics students.

Supports foundational understanding in quantum theory.

Abstract

In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students than the formal abstract one.