Hilbert Space

Inner Product Spaces^{inner product space} and Hilbert Spaces^{Hilbert space} are covered.

Inner Product Spaces

• What is an Inner Product Space?
  • Properties of Norms Associated with Inner Products
  • The Inner Product is a Continuous Mapping
  • The Relationship among Inner Product Spaces, Normed Spaces, and Metric Spaces
• Cauchy-Schwarz Inequality
• Properties of 0 in Inner Product Spaces
• Orthogonality, Orthogonal Sets, Orthonormal Sets
• Fourier Coefficients, Fourier Series

Hilbert Spaces

• Hilbert Space
• Theorem of the Shortest Vector
• Orthogonal Decomposition Theorem
• Hilbert Spaces are Reflexive
• Weak Convergence

Functionals

• Riesz Representation Theorem

Linear Operators

• Adjoint Operator
• Properties of the Adjoint Operator
• Orthogonal Projection
• Adjoint Operator from Hilbert Space to $\ell^{2}$ Space

Bessel Sequences

• Bessel Sequence
• Bessel Sequences in Hilbert Spaces with Dense Subspaces
• Generalized Bessel Inequality

Orthogonal Bases

• Reordering of Vector Spaces
• Gram-Schmidt Orthogonalization in Separable Hilbert Spaces
• Separable Hilbert Spaces are Isometrically Isomorphic to $\ell^{2}$ Space
• Orthonormal Bases and Unitary Operators
• Riesz Basis
• Frames

Function Spaces

• Various Function Spaces
• The Support of Functions and Classes of Continuous Function Spaces
• Hölder Continuous Function Space

References

• Walter Rudin, Real and Complex Analysis (3rd Edition, 1987)
• Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications (2nd Edition, 1999)
• Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003)
• Ole Christensen, Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering (2010)