Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
Introduction to Banach and Hilbert Spaces  Part I

Jan Popiolek

Warsaw University, Bialystok
Summary.

Basing on the notion of real linear space (see [11])
we introduce real unitary space.
At first, we define the scalar product of two vectors
and examine some of its properties. On the base of this notion we
introduce the norm and the distance in real unitary space and study
properties of these concepts.
Next, proceeding from the definition of the sequence
in real unitary space and basic operations on sequences we prove
several theorems which will be used in our further considerations.
MML Identifier:
BHSP_1
The terminology and notation used in this paper have been
introduced in the following articles
[4]
[12]
[1]
[9]
[5]
[2]
[3]
[13]
[8]
[6]
[11]
[10]
[7]
Contents (PDF format)
Bibliography
 [1]
Grzegorz Bancerek.
The ordinal numbers.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [3]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Jan Popiolek.
Some properties of functions modul and signum.
Journal of Formalized Mathematics,
1, 1989.
 [7]
Jan Popiolek.
Real normed space.
Journal of Formalized Mathematics,
2, 1990.
 [8]
Andrzej Trybulec.
Domains and their Cartesian products.
Journal of Formalized Mathematics,
1, 1989.
 [9]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
 [10]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
Journal of Formalized Mathematics,
1, 1989.
 [11]
Wojciech A. Trybulec.
Vectors in real linear space.
Journal of Formalized Mathematics,
1, 1989.
 [12]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [13]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received July 19, 1991
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