# The Hahn-Banach Theorem: a proof of the equivalence between the analitic and geometric versions

## TL;DR

This paper provides a straightforward proof demonstrating the equivalence between the analytic and geometric versions of the Hahn-Banach Theorem in the real case, and discusses extensions to the complex case.

## Contribution

It offers a simple, direct proof of the equivalence between the analytic and geometric forms of the Hahn-Banach Theorem in real spaces.

## Key findings

- Established the equivalence between analytic and geometric versions in real spaces.
- Summarized known proofs of reciprocal implications and direct proofs.
- Extended the discussion to the complex case using real case results.

## Abstract

We present here a simple and direct proof of the classic geometric version of Hahn-Banach Theorem from its analitic version, in the real case. The reciprocal implication, and the direct proofs of both versions, are already well kown, but they are also summarized. For the complex case, in both versions the Hahn-Banach Theorem is deduced from the real case, as it is well known.