Functional Analysis
Academic (University) preparation with worked examples, written practice and independent learning assessments.
Independent university-level study of Functional Analysis. This original curriculum draws on public academic subject descriptions; it is not an official faculty syllabus, university credit, accreditation or an Ivy League qualification. Topics covered: Normed spaces; Banach spaces; Linear operators; Operator norms. Hilbert spaces; Orthogonality; Projection theorem; Riesz representation. Dual spaces; Hahn-Banach theorem; Weak topology; Weak convergence. Uniform boundedness; Open mapping theorem; Closed graph theorem; Applications. Compact operators; Spectral theory; Fredholm alternative; Integral operators. Self-adjoint operators; Spectral measures; Unbounded operators; Domains. Distributions; Test functions; Weak derivatives; Sobolev spaces. Variational problems; Operator equations; PDE connections; Functional analytic proofs. Projects use safe simulations, public datasets or textual analysis; specialist subject review is required. Includes lessons, supplementary YouTube tutorials, practical exercises, quizzes and free/paid practice tests. Erudex is independent, not affiliated with the source institution or awarding body. Its 80% learning checks are not official grades or university credit. Content requires qualified subject review before use as a complete teaching programme.
Curriculum
One complete track with its own final exam and certificate.
Independent university-level study of Functional Analysis. This original curriculum draws on public academic subject descriptions; it is not an official faculty syllabus, university credit, accreditation or an Ivy League qualification. Topics covered: Normed spaces; Banach spaces; Linear operators; Operator norms. Hilbert spaces; Orthogonality; Projection theorem; Riesz representation. Dual spaces; Hahn-Banach theorem; Weak topology; Weak convergence. Uniform boundedness; Open mapping theorem; Closed graph theorem; Applications. Compact operators; Spectral theory; Fredholm alternative; Integral operators. Self-adjoint operators; Spectral measures; Unbounded operators; Domains. Distributions; Test functions; Weak derivatives; Sobolev spaces. Variational problems; Operator equations; PDE connections; Functional analytic proofs. Projects use safe simulations, public datasets or textual analysis; specialist subject review is required.
Normed spaces; Banach spaces; Linear operators; Operator norms
- Foundations of Normed Vector Spaces and Topological Equivalences Lab60 min
- Banach Spaces, Completeness, and Space Completions Lab60 min
- Bounded Linear Operators and the Space B(X, Y) Lab60 min
- Operator Norms, Spectral Properties, and Invertibility Lab60 min