# Short linear maps

## Idea

The short linear maps are the morphisms of the category Ban of Banach spaces (or its full subcategory Hilb of Hilbert spaces), at least if one wishes to recover isometric isomorphism?s as the abstract isomorphisms of this category.

## Definition

Given Banach spaces $V$ and $W$, a short linear map form $V$ to $W$ is a (total) function $f\colon V \to W$ that is both short and linear, equivalently a bounded linear operator whose norm is at most $1$. These are also called contractive (or weakly contractive) linear maps; they may also be called short (or contractive or weakly contractive) operators (with linearity usually assumed but sometimes mentioned), or even simply (weak) contractions.

## On Hilbert spaces

Short linear maps from a Hilbert space to itself (that is endomorphisms in Hilb) are discussed (as simply ‘contractions’) in operator theory?; see Wikipedia.

Created on August 16, 2012 19:33:23 by Toby Bartels (98.19.44.121)