Hilbert Space
Inner Product Spacesinner product space and Hilbert SpacesHilbert space are covered.
Inner Product Spaces
- What is an Inner Product Space?
- 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
Linear Operators
- Adjoint Operator
- Properties of the Adjoint Operator
- Orthogonal Projection
- Adjoint Operator from Hilbert Space to $\ell^{2}$ Space
Bessel Sequences
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)
All posts
- Hilbert Spaces in Functional Analysis
- Proof of the Shortest Vector Theorem
- Orthogonal Decomposition Theorem Proof
- Proof of Liouville's Theorem
- Hilbert Spaces are Reflexive: A Proof
- Holder Continuous Function Spaces
- Hilbert Space Adjoint Operators
- Orthogonal Projection in Hilbert Spaces
- Hilbert Space to L2 Space: The Adjoint Operator
- Lies Basis
- Hilbert Space Bessel Sequences
- Proof of the Generalized Bessel's Inequality in Hilbert Spaces
- Bessel Sequences in Hilbert Spaces with Dense Subspaces
- Reordering of Vector Spaces
- Gram-Schmidt Orthogonalization in Separable Hilbert Spaces
- Proving that Every Separable Hilbert Space is Isometric to the l^2 space
- Hilbert Space's Orthonormal Basis and Unitary Operator
- The Support of Functions and the Class of Continuous Function Spaces
- Hilbert Space Frames
- Weak Convergence in Hilbert Spaces
- Relations among Inner Product Spaces, Normed Spaces, and Metric Spaces
- Inner product spaces
- Inner Product Spaces and the Cauchy-Schwarz Inequality
- Properties of the Norm Associated with the Inner Product Defined in Inner Space
- Orthogonality, Orthogonal Sets, and Orthonormal Sets in Inner Product Spaces
- Properties of Zero in Inner Product Spaces
- Properties of Adjoint Operators
- Inner Product is a Continuous Mapping
- Generalized Fourier Coefficients and Fourier Series in Hilbert Spaces
- Various Function Spaces
- L² Space