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Contents

• V Full version
• 0 Introduction
• 1 Normed vector spaces
• 1.1 Bounded linear maps
• 1.2 Dual spaces
• 1.3 Adjoint
• 1.4 The double dual
• 1.5 Isomorphism
• 1.6 Finite-dimensional normed vector spaces
• 1.7 Hahn–Banach Theorem
• 2 Baire category theorem
• 2.1 The Baire category theorem
• 2.2 Some applications
• 3 The topology of C(K)
• 3.1 Normality of compact Hausdorff spaces
• 3.2 Tietze-Urysohn extension theorem
• 3.3 Arzela-Ascoli theorem
• 3.4 Stone–Weierstrass theorem
• 4 Hilbert spaces
• 4.1 Inner product spaces
• 4.2 Riesz representation theorem
• 4.3 Orthonormal systems and basis
• 4.4 The isomorphism with ℓ₂
• 4.5 Operators
• 4.6 Self-adjoint operators