Di dalam tesis ini, dibicarakan topologi seminorm, topologi lemah sigma(X,X^*) pada ruang vektor topologis X dan topologi lemah* sigma(X^*, X) pada ruang dualnya. Topologi lemah sigma(X,X^*) pada X merupakan topologi yang dibangun keluarga seminorm {pf :f \in X^*} dengan pf (x) = |f (x)| , x \in X. Topologi lemah* sigma(X^*, X) merupakan topologi konveks lokal sehingga (X^*,sigma(X^*, X)) ruang konveks lokal. Selanjutnya, jika X ruang konveks lokal maka sigma(X,X^*) juga ruang konveks lokal. Di samping itu, di dalam tesis ini diperkenalkan kelas baru dari ruang konveks lokal yaitu ruang barrelled dan sub-kelasnya yaitu ruang countably barrelled. Untuk mengkaji ruang barrelled dan countably barrelled dibutuhkan topologi lemah pada ruang konveks lokal dan topologi lemah* pada dualnya. Kata kunci: seminorm, topologi lemah, ruang konveks lokal, ruang barrelled.

In this thesis, we study the seminorm topology, the weak topology sigma(X,X^*) on topological vector spaces X and the weak* topology sigma(X^*, X) on their duals. The weak topology sigma(X,X^*) on X is the seminorm topology induced by the family of seminorms {pf :f \in X^*} with pf (x) = |f (x)| , x \in X. The weak* topology sigma(X^*, X) is a locally convex topology. Therefore, (X^*,sigma(X^*, X)) is a locally convex space. Next, if X locally convex space then sigma(X,X^*) is a locally convex space. In addition, in this thesis, we also introduce a new class of locally convex spaces that is barrelled spaces and their subclasses that are countably barrelled spaces. We need the weak topology on locally convex spaces and the weak* topology on their duals to study barrelled spaces and countably barrelled spaces. Keywords: seminorm, weak topology, locally convex spaces, barreled spaces.

Kata Kunci : seminorm, topologi lemah, ruang konveks lokal, ruang barrelled