Banach + Hilbert spaces; bounded + unbounded operators; spectral theory; distributions; Banach algebras; locally-convex spaces; weak topologies.
functional-analysis
Hahn-Banach extension theorem
Hahn 1927 / Banach 1929: bounded linear functional on subspace extends to whole Banach space preserving norm. Foundational. Proof via…
Banach-Steinhaus uniform boundedness
Pointwise-bounded family of bounded operators on Banach space is uniformly-bounded in operator-norm. Baire-category argument. Foundational…
Open mapping + closed graph
Open-mapping theorem: surjective bounded operator between Banach spaces is open. Closed-graph: linear operator with closed graph between…
Spectral theorem for bounded self-adjoint operators
Bounded self-adjoint operator A on Hilbert space H has integral representation A = ∫λ dE(λ) over spectral resolution E. Generalises…
Riesz representation theorem
Continuous linear functional on Hilbert space H given by inner-product with unique element y ∈ H. Generalises to dual-spaces of L^p / C(K)…
Compact operators & Fredholm theory
Compact operators K: B(X)→B(X) (image of unit-ball precompact). Fredholm-alternative for I-K (cross-listed L0 integral-eq). Spectrum of…
Distributions (Schwartz)
Schwartz 1950 'Théorie des distributions': continuous linear functionals on test-function space D(R^n). Generalises functions; allows…
Sobolev spaces + embedding
W^{k,p}(Ω) functions with k weak-derivatives in L^p. Sobolev embedding W^{k,p} ⊂ L^q for appropriate q (or C^m for k > n/p). Foundation of…
Fréchet + locally convex spaces
Locally-convex topological-vector-space; Fréchet = complete metrizable LCS. Generalises Banach. Examples: smooth-functions C∞ / Schwartz S…
C*-algebra + GNS construction
C*-algebra: Banach-algebra with involution * + ||a*a||=||a||². Gelfand-Naimark-Segal: every C*-algebra embeds in B(H) for some Hilbert H.…
Banach algebras (Gelfand theory)
Commutative-Banach-algebra A: continuous-spectrum = maximal-ideals = characters. Gelfand-transform A → C(M(A)). Wiener-tauberian theorem…
Weak topologies + Banach-Alaoglu
Weak-* topology on dual X*; Banach-Alaoglu: closed-unit-ball of X* is weak-*-compact. Foundation of variational + optimisation methods.…
Hahn-Banach extension theorem (1929)
Hahn 1927 / Banach 1929: bounded linear functional on subspace extends to full space without increasing norm; foundation of dual-space…
Uniform boundedness (Banach-Steinhaus)
Banach-Steinhaus 1927: pointwise-bounded family of bounded linear operators on Banach space is uniformly bounded; consequence of Baire…
Open mapping theorem (Banach 1929)
Banach 1929 open-mapping theorem: surjective bounded linear operator between Banach spaces is open; implies inverse is bounded if injective.
Closed graph theorem
Linear operator T:X->Y between Banach spaces is bounded iff its graph is closed in XxY; equivalent to open-mapping + Hahn-Banach.
Riesz representation theorem (L2)
Every continuous linear functional on Hilbert space H represented as inner product <.,y> for unique y in H; bridges algebraic + topological…
Spectral theorem (self-adjoint)
Every bounded self-adjoint operator on Hilbert space has spectral resolution as integral against projection-valued measure on its real…
Banach (1932)
S Banach 1932 'Théorie des Opérations Linéaires'; modern modern foundational text + Banach-Tarski + 2024 Banach-spaces structure-theory.
Hahn-Banach (1929)
H Hahn 1927 + S Banach 1929 + Helly 1921 Hahn-Banach-extension; modern modern foundational text + dual-spaces + reflexive-Banach.
Banach-Steinhaus (1927)
S Banach-H Steinhaus 1927 uniform-boundedness; modern modern foundational text + 2024 Baire-category foundation operator-bounds.
Riesz representation (1907)
F Riesz 1907 + 1909 representation-theorem; modern modern foundational text + Hilbert-space + 2024 Lebesgue-Stieltjes integral.
Spectral theorem (von Neumann 1929)
J von Neumann 1929-1932 spectral-theorem self-adjoint; modern modern foundational text + 2024 unbounded operators + Stone's theorem 1932.
C*-algebra (Gelfand-Naimark 1943)
I Gelfand-M Naimark 1943 + Segal 1947 GNS construction; modern modern foundational text + 2024 noncommutative-geometry Connes IM Beilinson.