Hilbert Space Orthonormal Basis

A Hilbert Space Orthonormal Basis is a vector space basis that can be used to create hilbert space representation systems (that support hilbert space vector expansion tasks).

• AKA: Hilbert Basis, Hilbert Space Basis.
• Context:
  • It can typically represent Hilbert Space Vectors through hilbert space infinite series expansions with hilbert space fourier coefficients.
  • It can typically satisfy Hilbert Space Orthonormality Condition through hilbert space inner product relationships.
  • It can typically enable Hilbert Space Parseval Identity through hilbert space norm preservation.
  • It can typically provide Hilbert Space Completeness Property through hilbert space dense linear span.
  • ...
  • It can often support Hilbert Space Fourier Analysis through hilbert space harmonic decomposition.
  • It can often enable Hilbert Space Quantum State Representation through hilbert space state vector expansion.
  • It can often implement Hilbert Space Function Approximation through hilbert space basis expansion.
  • ...
  • It can range from being a Finite Hilbert Space Orthonormal Basis to being a Countably Infinite Hilbert Space Orthonormal Basis, depending on its hilbert space basis cardinality.
  • It can range from being a Separable Hilbert Space Orthonormal Basis to being a Non-Separable Hilbert Space Orthonormal Basis, depending on its hilbert space separability.
  • ...
  • It can generate Hilbert Space Coordinate System through hilbert space basis coefficients.
  • It can ensure Hilbert Space Unique Representation via hilbert space orthogonal decomposition.
• Examples:
  • Hilbert Space Orthonormal Basis Types, such as:
    • Hilbert Space Finite Orthonormal Basises, such as:
      • Hilbert Space Standard Basis for hilbert space finite dimensional euclidean space.
      • Hilbert Space Quantum Spin Basis for hilbert space finite quantum system.
    • Hilbert Space Infinite Orthonormal Basises, such as:
      • Hilbert Space Fourier Basis for hilbert space periodic function space.
      • Hilbert Space Wavelet Basis for hilbert space multiresolution analysis.
  • Hilbert Space Orthonormal Basis Applications, such as:
    • Hilbert Space Function Space Basises, such as:
      • Hilbert Space Legendre Polynomial Basis for hilbert space square integrable function.
      • Hilbert Space Hermite Function Basis for hilbert space quantum harmonic oscillator.
    • Hilbert Space Quantum Mechanics Basises, such as:
      • Hilbert Space Energy Eigenstate Basis for hilbert space hamiltonian diagonalization.
  • ...
• Counter-Examples:
  • Vector Space Basis, which uses finite linear combinations rather than hilbert space infinite series expansion.
  • Linearly Independent Set, which provides vector independence without hilbert space orthonormality requirement.
  • Vector Space Spanning Set, which covers vector space without hilbert space inner product orthogonality.
• See: Hilbert Space, Vector Space Basis, Inner Product Space, Mathematical Space, Euclidean Vector Space.

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