Functional Analysis
A rigorous course in functional analysis, the study of infinite-dimensional vector spaces of functions and sequences together with the linear operators that act on them, built around the question of which parts of linear algebra survive in infinite dimensions: norms and the equivalence of all norms in finite dimensions, Banach spaces with complete proofs that the sequence spaces ℓᵖ and c₀ and the space of continuous functions are complete, the series criterion for completeness and Riesz’s lemma showing that unit balls of infinite-dimensional spaces are never compact, bounded linear operators with operator norms of matrices and integral operators, the Neumann series and Fredholm integral equations, dual spaces with the identification of the duals of ℓᵖ and c₀, Hilbert spaces with the Cauchy–Schwarz inequality, the parallelogram law and the projection theorem, orthonormal bases with Bessel’s inequality, Parseval’s identity, the Legendre polynomials and the completeness of the trigonometric system with the Gibbs phenomenon, the Riesz representation theorem with a reproducing kernel in a Sobolev space and adjoint, self-adjoint, unitary and normal operators, the Hahn–Banach theorem in its analytic and geometric forms with norming functionals, separating hyperplanes, the bidual and Banach limits, the Baire category theorem with the uniform boundedness principle, continuous functions with divergent Fourier series, the open mapping and closed graph theorems and the Hellinger–Toeplitz theorem, weak and weak-star convergence with Mazur’s theorem, the Banach–Alaoglu theorem, reflexive spaces and the direct method of the calculus of variations, compact and Hilbert–Schmidt operators with the Fredholm alternative and the index of the shift, spectra of bounded operators and the spectral theorem for compact self-adjoint operators applied to the vibrating string, and a capstone on unbounded self-adjoint operators, momentum and its self-adjoint extensions, the particle in a box, the Heisenberg uncertainty principle, and weak solutions of boundary value problems through the Poincaré inequality, the Lax–Milgram theorem and Galerkin’s finite element method, with complete proofs, worked examples and runnable Python illustrations throughout.
MATH-445 Year 4 3 credits 12 lessons · 4 h 44 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
This is the American version of this course. See the Canadian version.
What you'll learn
- Explain which results of finite-dimensional linear algebra fail in infinite dimensions, and prove that all norms on a finite-dimensional space are equivalent.
- Prove completeness of the sequence spaces ℓᵖ, c₀ and the function space C[a, b], use the absolutely convergent series criterion, and apply Riesz’s lemma to show that unit balls of infinite-dimensional spaces are not compact.
- Characterize bounded linear operators, compute operator norms of matrices and integral operators, invert perturbations of the identity by the Neumann series, and identify the duals of ℓᵖ and c₀.
- Use the Cauchy–Schwarz inequality and the parallelogram law, prove the closest-point and projection theorems in Hilbert space, and solve least-squares problems through the normal equations.
- Prove Bessel’s inequality and Parseval’s identity, characterize orthonormal bases, construct the Legendre polynomials, and prove that the trigonometric system is an orthonormal basis of L²(−π, π).
- Prove the Riesz representation theorem, compute representers of functionals in L² and Sobolev spaces, and construct and use adjoints of bounded operators, including the relation between kernels of adjoints and ranges.
- Prove the Hahn–Banach theorem in its analytic and geometric forms, and use it to construct norming functionals, separating hyperplanes, the canonical embedding into the bidual and Banach limits.
- Prove the Baire category theorem, the uniform boundedness principle, the open mapping theorem and the closed graph theorem, and apply them to divergent Fourier series and to symmetric operators.
- Work with weak and weak-star convergence, prove the sequential Banach–Alaoglu theorem, determine which classical spaces are reflexive, and apply weak compactness to convex minimization.
- Prove that compact operators form a closed ideal, show that Hilbert–Schmidt integral operators are compact, prove the Fredholm alternative, and compute Fredholm indices of shift operators.
- Compute spectra of bounded operators, prove the spectral theorem for compact self-adjoint operators, and use it to diagonalize Green’s operators and integral operators with symmetric kernels.
- Distinguish symmetric and self-adjoint unbounded operators, derive quantized energies and the uncertainty principle, and obtain weak solutions of boundary value problems with the Riesz and Lax–Milgram theorems and Galerkin’s method.
Course content
12 lessons · 4 h 44 min total · final exam
- 1 Why functional analysis: linear algebra in infinite dimensions 23 min · quiz
- 2 Normed and Banach spaces 23 min · quiz
- 3 Bounded linear operators and dual spaces 23 min · quiz
- 4 Hilbert spaces, orthogonality and projections 22 min · quiz
- 5 Orthonormal bases and Fourier series 23 min · quiz
- 6 The Riesz representation theorem and adjoints 22 min · quiz
- 7 The Hahn–Banach theorem 23 min · quiz
- 8 Baire category, the uniform boundedness principle and the open mapping theorem 23 min · quiz
- 9 Weak topologies and reflexivity 24 min · quiz
- 10 Compact operators and Fredholm theory 25 min · quiz
- 11 Spectral theory of compact self-adjoint operators 25 min · quiz
- 12 Capstone: unbounded operators, quantum mechanics and partial differential equations 28 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A rigorous course in functional analysis, the study of infinite-dimensional vector spaces of functions and sequences together with the linear operators that act on them, built around the question of which parts of linear algebra survive in infinite dimensions: norms and the equivalence of all norms in finite dimensions, Banach spaces with complete proofs that the sequence spaces ℓᵖ and c₀ and the space of continuous functions are complete, the series criterion for completeness and Riesz’s lemma showing that unit balls of infinite-dimensional spaces are never compact, bounded linear operators with operator norms of matrices and integral operators, the Neumann series and Fredholm integral equations, dual spaces with the identification of the duals of ℓᵖ and c₀, Hilbert spaces with the Cauchy–Schwarz inequality, the parallelogram law and the projection theorem, orthonormal bases with Bessel’s inequality, Parseval’s identity, the Legendre polynomials and the completeness of the trigonometric system with the Gibbs phenomenon, the Riesz representation theorem with a reproducing kernel in a Sobolev space and adjoint, self-adjoint, unitary and normal operators, the Hahn–Banach theorem in its analytic and geometric forms with norming functionals, separating hyperplanes, the bidual and Banach limits, the Baire category theorem with the uniform boundedness principle, continuous functions with divergent Fourier series, the open mapping and closed graph theorems and the Hellinger–Toeplitz theorem, weak and weak-star convergence with Mazur’s theorem, the Banach–Alaoglu theorem, reflexive spaces and the direct method of the calculus of variations, compact and Hilbert–Schmidt operators with the Fredholm alternative and the index of the shift, spectra of bounded operators and the spectral theorem for compact self-adjoint operators applied to the vibrating string, and a capstone on unbounded self-adjoint operators, momentum and its self-adjoint extensions, the particle in a box, the Heisenberg uncertainty principle, and weak solutions of boundary value problems through the Poincaré inequality, the Lax–Milgram theorem and Galerkin’s finite element method, with complete proofs, worked examples and runnable Python illustrations throughout.
Part of the Science stream, typically taken in year 4.