Kekonvergenan Lemah pada Ruang Hilbert (Jurnal EurekaMatika, 2017)
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.ABSTRAKPenelitian ini mengkaji mengenai kekonvergenan lemah padaruang Hilbert atas lapangan real. Kekonvergenan lemah termotivasi olehkekonvergenan kuat sehingga terdapat beberapa sifat dari kekonvergenankuat yang berlaku pada kekonvergenan lemah seperti ketunggalan limit,kelinearan limit, dan keterbatasan suatu barisan. Keterkaitan antarakonvergen kuat dan lemah mengakibatkan terdapat pendefinisian dan sifat- sifat dari barisan Cauchy lemah dan himpunan kompak secara barisan dansecara lemah. Di akhir pembahasan dibicarakan mengenai keberlakuanTeorema Bolzano-Weierstrass pada ruang Hilbert.
Keywords: Hilbert Space; Strong Convergence; Weak Convergence; Weakly Compact Set; Bolzano-Weierstrass Theorem