Hölder’s and Minkowski’s inequalities. Completeness and separability of known sequence spaces and function spaces.  Normed spaces. Banach spaces. Finite-dimensional spaces. Operators and functionals.
Dual Banach spaces. Reflectivity. Hahn-Banach theorem and its applications. Applications of Baire’s theorem: The open mapping  theorem. The uniform bound principle, the closed graph theorem.
Inner-product spaces. Hilbert spaces. Orthogonal and orthonormal systems. Orthogonal complement and projections. The Riesz representation theorem. Adjoint, self-adjoint, normal, isometry, unitary, compact and projection operators.
Resolvent set and spectrum. The spectrum of adjoint and compact operators. Solution of equations in Hilbert spaces.