# On the duals of normed spaces and quotient shapes

## TL;DR

This paper explores properties of the dual functor in normed spaces, revealing how it affects embeddings, dimensions, and shape classifications, leading to new extension theorems.

## Contribution

It provides a comprehensive classification of finite quotient shapes of normed spaces and analyzes the dual functor's impact on these properties.

## Key findings

- D turns canonical embeddings into retractions
- D increases algebraic dimension only
- D preserves finite quotient shape type

## Abstract

Some properties of the (normed) dual Hom-functor $D$ and its iterations $D^n$ are exhibited. For instance: $D$ turns every canonical embedding (in the second dual space) into a retraction (of the third dual onto the first one); $D$ rises the countably infinite (algebraic) dimension only; $D$ does not change the finite quotient shape type. By means of that, the finite quotient shape classification of normed vectorial spaces is completely solved. As a consequence, two extension type theorems are derived.