Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003 Association of Mizar Users

The Inner Product of Finite Sequences and of Points of $n$-dimensional Topological Space


Kanchun
Shinshu University, Nagano
Yatsuka Nakamura
Shinshu University, Nagano

Summary.

First, we define the inner product to finite sequences of real value. Next, we extend it to points of $n$-dimensional topological space ${\calE}^{n}_{\rmT}$. At the end, orthogonality is introduced to this space.

MML Identifier: EUCLID_2

The terminology and notation used in this paper have been introduced in the following articles [12] [3] [1] [10] [8] [2] [4] [7] [9] [5] [6] [11]

Contents (PDF format)

  1. Preliminaries
  2. Inner Product of Finite Sequences
  3. Inner Product of Points of ${\calE}^{n}_{\rmT}$

Bibliography

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[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[3] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[5] Czeslaw Bylinski. The sum and product of finite sequences of real numbers. Journal of Formalized Mathematics, 2, 1990.
[6] Agata Darmochwal. The Euclidean space. Journal of Formalized Mathematics, 3, 1991.
[7] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[8] Jan Popiolek. Some properties of functions modul and signum. Journal of Formalized Mathematics, 1, 1989.
[9] Agnieszka Sakowicz, Jaroslaw Gryko, and Adam Grabowski. Sequences in $\calE^N_\rmT$. Journal of Formalized Mathematics, 6, 1994.
[10] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[11] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.
[12] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

Received February 3, 2003


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