Parseval identity for orthonormal systems in L^2

Target entity: Parseval identity for orthonormal systems in L^2
Context triple: [Riesz–Fischer theorem, implies, Parseval identity for orthonormal systems in L^2]

• A. Paley–Wiener theorem
  The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.
• B. Dirichlet theorem on Fourier series
  The Dirichlet theorem on Fourier series gives conditions under which a periodic function can be represented by a convergent Fourier series, specifying how and where the series converges to the function.
• C. Littlewood–Paley theory
  Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
• D. Carleson theorem on almost-everywhere convergence
  The Carleson theorem on almost-everywhere convergence is a fundamental result in harmonic analysis stating that the Fourier series of any square-integrable function on the circle converges almost everywhere to the function itself.
• E. Riesz basis
  A Riesz basis is a sequence in a Hilbert space that is complete and behaves like an orthonormal basis up to a bounded, invertible linear transformation, allowing stable and unique expansions of vectors.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Parseval identity for orthonormal systems in L^2
Target entity description: The Parseval identity for orthonormal systems in \(L^2\) is a fundamental result in functional analysis stating that the squared norm of a function equals the sum of the squares of its Fourier (or orthonormal expansion) coefficients, expressing energy conservation between a function and its orthonormal series representation.

• A. Paley–Wiener theorem
  The Paley–Wiener theorem is a fundamental result in harmonic analysis that characterizes which functions arise as Fourier transforms of compactly supported functions (or distributions), linking analytic properties of entire functions with support properties in the original domain.
• B. Dirichlet theorem on Fourier series
  The Dirichlet theorem on Fourier series gives conditions under which a periodic function can be represented by a convergent Fourier series, specifying how and where the series converges to the function.
• C. Littlewood–Paley theory
  Littlewood–Paley theory is a collection of techniques in harmonic analysis that decompose functions into frequency-localized pieces to study their behavior in L^p spaces and related function spaces.
• D. Carleson theorem on almost-everywhere convergence
  The Carleson theorem on almost-everywhere convergence is a fundamental result in harmonic analysis stating that the Fourier series of any square-integrable function on the circle converges almost everywhere to the function itself.
• E. Riesz basis
  A Riesz basis is a sequence in a Hilbert space that is complete and behaves like an orthonormal basis up to a bounded, invertible linear transformation, allowing stable and unique expansions of vectors.
• F. None of above. chosen