Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

## Hermitan Functionals. Canonical Construction of Scalar Product in Quotient Vector Space

Jaroslaw Kotowicz
University of Bialystok

### Summary.

In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The construction of scalar product in quotient vector space from non negative hermitan functions is the main result of the article.

This work has been partially supported by TRIAL-SOLUTION grant IST-2001-35447 and SPUB-M grant of KBN.

#### MML Identifier: HERMITAN

The terminology and notation used in this paper have been introduced in the following articles [18] [5] [23] [1] [19] [7] [8] [17] [3] [2] [21] [12] [24] [4] [20] [6] [9] [22] [14] [15] [11] [10]

#### Contents (PDF format)

1. Auxiliary Facts about Complex Numbers
2. Antilinear Functionals in Complex Vector Spaces
3. Sesquilinear Forms in Complex Vector Spaces
4. Kernel of Hermitan Forms and Hermitan Forms in Quotient Vector Spaces
5. Scalar Product in Quotient Vector Space Generated by Nonnegative Hermitan Form

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