On Entropy-Driven Hemi-Inner Products
Achim Schlather, Martin Schlather, Jannik Bochynek · Preprints.org · 2025
It is well known that the inner product defines the norm in a Hilbert space, and conversely, the norm determines the inner product via the polarization identity. Since both norms and entropy functions are non-negative, one may formally define a weak form of an entropy-induced inner product by means of the polarization identity. We present several identities traditionally associated with genuine inner products that continue to hold in a more general setting. In conclusion, an entropy function can endow a simple algebraic structure with properties reminiscent of Hilbert spaces. We consider various examples and show that an entropy can be reconstructed from a weak inner product in case of cancellative commutative semigroups.