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(Semi-)norms, Banach spaces:
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L^p-space
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completion of a normed vector space
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noncompactness of the closed unit ball in infinite dimensions
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Inner products, Hilbert spaces:
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Cauchy-Schwarz inequality
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parallelogram rule
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orthonormal bases
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Parseval's identity
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Bessel's inequality
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classification of Hilbert spaces
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separability
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Fourier series
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Linear operators:
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bounded linear maps
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Neumann series
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Open Mapping Theorem (Banach-Schauder)
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Closed Graph Theorem
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Uniform Boundedness Theorem (Banach-Steinhaus)
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Dual spaces:
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Fréchet-Riesz representation theorem
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Hahn-Banach Theorem
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bidual space and reflexivity
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dual operator
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Operators on Hilbert spaces:
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adjoint of an operator
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spectrum
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spectral radius
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spectra of self-adjoint, unitary, normal, and compact operators
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Riesz-Schauder Theorem about a compact perturbation of the identity operator
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Spectral Theorem for a compact normal operator
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Fredholm alternative
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