本課程上接實變函數論，為上下兩學期各三學分的課。上學期的內容包含有拓樸向量空間，下學期則包含有算子理論及算子代數的基本理論。

近代分析所考慮的空間為函數空間（function spaces）。即為先固定一定義域，再將定義域上同類型（如同為連續、可微或可積）之函數集合起來，稱之為函數空間。在這個空間上，我們將探討它的代數性質及拓樸性質。上述之函數空間，我們統稱為拓樸向量空間。然後再進一步探討定義域及值域皆為拓樸向量空間的連續線性函數（我們稱之為算子）之性質，這已經進入到算子理論的領域了。接著再把定義在同一空間之所有連續線性函數集合起來即形成一算子代數。當然這部分授課的多寡將依時間而定。

以下為授課內容之細節，包含有：

Fall :

1. Hilbert space (orthogonality, Riesz representation theorem, bases)
2. Operators on Hilbert space (hermitian operator, normal operator, unitary operator, projection, compact operator)
3. Banach spaces (Hahn-Banach theorem, open mapping theorem, inverse mapping theorem, closed graph theorem, principle of uniform boundedness)
4. Locally convex spaces (seminorm, metrizable LCS, normable LCS, finite-dimensional LCS, separation theorem)

Spring :

1. Weak topologies (duality, Alaoglu's theorem, reflexivity, Krein-Milman theorem, Stone-Weierstrass theorem)
2. Linear operators on a Banach space (adjoint, Banach-Stone theorem, Schauder's theorem, Arzela-Ascoli theorem, weakly compact operator)
3. Banach algebras and Spectral theory (spectrum, Riesz functional calculus, spectral mapping theorem, Fredholm alternative, Gelfand-Mazur theorem, maximal ideal space, Gelfand transform)
4. C*-algebras (functional calculus, positive element, polar decomposition, representation, GNS construction)