<![CDATA[ABSTRAK. Penelitian 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 sifatsifatdaribarisanCauchylemahdanhimpunankompaksecarabarisandansecaralemah. Di akhir pembahasan dibicarakan mengenai keberlakuanTeorema Bolzano-Weierstrass pada ruang Hilbert.Kata kunci: Ruang Hilbert, konvergen kuat, konvergen lemah, himpunankompak secara barisan dan secara lemah, teorema Bolzano-Weiertrass.ABSTRACT. This study discusses the weak of convergence in Hilbertspace over the real field. The weak of convergence is motivated by a strongconvergence as a result that there are some properties of the strongconvergence which is applicable in the weak of convergence such asuniqueness of limit, linearity of limit, and boundedness of a sequence. Therelationship between strong and weak convergent implies that there are thedefinition and properties of weak Cauchy sequence and weakly compactset. In the end of the discussion discussed about the generalize of BolzanoWeierstrasstheoreminHilbertspace.Keywords: Hilbert Space, strong convergence, weak convergence, weaklycompact set, Bolzano-Weierstrass Theorem.]]>
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