SISTEM ORTONORMAL DALAM RUANG HILBERT
Hilbert space is one of the important inventions in mathematics. Historically, the theory of Hilbert space originated from David Hilbert’s work on quadratic form in infinitely many variables with their applications to integral equations. This paper contains some definitions such as vector space, normed space and inner product space (also called pre-Hilbert space), and which is important to construct the Hilbert space. The fundamental ideas and results are discussed with special attention given to finite dimensional pre-Hilbert space and some basic propositions of orthonormal systems in Hilbert space. This research found that each finite dimensional pre- Hilbert space is a Hilbert space. We have provided that a finite orthonormal systems in a Hilbert space X is complete if and only if this orthonormal systems is a basis of X.
Kumar, Nimit. Learning in Hilbert Spaces. Departement of Electrical Engineering at Indian Institute of
Technology, Kanpur. (2004).
Kreyszig, Erwin. Introductory Functional Analysis With Applications. John Wiley & Sons, New York. (1978).
Steven, Leon J. Aljabar Linear dan Aplikasinya. Erlangga, Jakarta. (2001).
Zeidler, Eberhard. Applied Functional Analysis, Application to Mathematical Physics. Springer-Verlag. (1995).