(untitled) (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. 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.