Kekonvergenan Lemah pada Ruang Hilbert
Keywords:
Hilbert Space, Strong Convergence, Weak Convergence, Weakly Compact Set, Bolzano-Weierstrass TheoremAbstract
This study discusses the weak of convergence in Hilbert space over the real field. The weak of convergence is motivated by a strong convergence as a result that there are some properties of the strong convergence which is applicable in the weak of convergence such as uniqueness of limit, linearity of limit, and boundedness of a sequence. The relationship between strong and weak convergent implies that there are the definition and properties of weak Cauchy sequence and weakly compact set. In the end of the discussion discussed about the generalize of Bolzano- Weierstrass theorem in Hilbert space.
ABSTRAK
Penelitian ini mengkaji mengenai kekonvergenan lemah pada
ruang Hilbert atas lapangan real. Kekonvergenan lemah termotivasi oleh
kekonvergenan kuat sehingga terdapat beberapa sifat dari kekonvergenan
kuat yang berlaku pada kekonvergenan lemah seperti ketunggalan limit,
kelinearan limit, dan keterbatasan suatu barisan. Keterkaitan antara
konvergen kuat dan lemah mengakibatkan terdapat pendefinisian dan sifat- sifat dari barisan Cauchy lemah dan himpunan kompak secara barisan dan
secara lemah. Di akhir pembahasan dibicarakan mengenai keberlakuan
Teorema Bolzano-Weierstrass pada ruang Hilbert.
References
H. Moscovich, “Norm, Strong, and Weak Operator Topologies on B (H),” 18 Mei 2009. [Online]. Available: u.math.biu.ac.il/~megereli/updated.pd. [Diakses 3 Oktober 2016].
E. Kreyzig, Introduction to Functional Analysis and Its Applications, New York: John Wiley and Sons, 1978.
A. L. Cauchy, Cours d'Analyse, 1821.
A. V. Balakrishnan, Applications of Mathematics: Applied Functional Analysis, New York: Springer-Verlag, 1981.
A. W. Roberts dan D. E. Varberg, Convex Functions, New York: Academic Press, Inc., 1973.
R. Mukti, “Skripsi Kekonvergenan Lemah Pada Ruang Vektor Bernorma,” tidak diterbitkan, Bandung, 2010.
D. H. Griffel, Applied Functional Analysis, New York: Dover Publications, Inc., 1981.
M. A. M. Fereirra, “The Weak Convergence in Hilbert Spaces Concept Contruction,” International Sciences and Computing, 2014.
R. G. Bartle., dan D. R. Sherbert, Introduction To Real Analysis, Urbana, Illinois, 2010.
H. Anton, Aljabar Libear Elementer (Edisi Lima), Jakarta: Erlangga, 1987.
