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In classical Analysis, Euclidean spaceSpaceEuclidean of dimension n refers to the n-fold Cartesian product \(\mathbb {R}^n=\mathbb {R}\times \dots \times \mathbb {R}\) endowed with its natural metric structure. This means that, inspired by Pythagoras’ theorem, one defines the lengthLengthvector of a vectorVectorin Eucledean space  in \(\mathbb {R}^n\) and hence the distance between two points in \(\mathbb {R}^n\) as the length of the segment joining them, the latter being just the “size” of the difference between the two vectors associated to the points.

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Structures and Functions on Euclidean Space

  • 章节

  • 发表日期：

    2025年

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  • Marco Baronti,
  • Enrico Calcagno,
  • Filippo De Mari,
  • Robertus van der Putten

摘要

In classical Analysis, Euclidean spaceSpaceEuclidean of dimension n refers to the n-fold Cartesian product \(\mathbb {R}^n=\mathbb {R}\times \dots \times \mathbb {R}\) endowed with its natural metric structure. This means that, inspired by Pythagoras’ theorem, one defines the lengthLengthvector of a vectorVectorin Eucledean space  in \(\mathbb {R}^n\) and hence the distance between two points in \(\mathbb {R}^n\) as the length of the segment joining them, the latter being just the “size” of the difference between the two vectors associated to the points.