Extens\~ao de aplica\c{c}\~oes na esfera de um espa\c{c}o vetorial com produto interno

TL;DR

This paper proves that any transformation of a sphere in a vector space with an inner product that preserves the inner product can be extended to a linear isometry of the entire space.

Contribution

It demonstrates that inner product-preserving maps on a sphere are restrictions of linear isometries in the vector space, extending previous understanding of such transformations.

Findings

Inner product-preserving maps on spheres are restrictions of linear isometries.

The paper provides a formal proof of this extension property.

It clarifies the structure of transformations preserving inner products in vector spaces.

Abstract

Being E a vector space with inner product and S the sphere of E, will be given a demonstration that every application of the sphere S itself it such that preserve inner product is the restriction of a linear isometry in E.