# Partial Inner Products on Antiduals

## TL;DR

This paper explores extending inner products from vector spaces to their antiduals, revealing that while full extensions lack weak continuity, partial extensions preserve key structures like Hilbert space completion.

## Contribution

It introduces the concept of partial inner product extensions on antiduals, highlighting their ability to retain important mathematical structures despite the absence of weak continuity.

## Key findings

- Partial extensions recover Hilbert space structures
- Full extensions are not weakly continuous
- Partial extensions maintain antiduality pairing

## Abstract

We discuss extensions of an inner product from a vector space to its full antidual. None of these extensions is weakly continuous, but partial extensions recapture some familiar structure including the Hilbert space completion and the antiduality pairing.