Alphabeta MathPersonal research
Exterior geometry and function spaces
Specify the obstacle and generalized-ray conventions, Sobolev and time-space notation, Dirichlet realization and spectral scales; connect bounded fractional inputs to exterior localization, traces, compactness and continuation.
1 · Prerequisites
- The L² operator associated with a symmetric elliptic form
- Hilbert space
- The associated elliptic operator is densely defined, symmetric and lower bounded
- The symmetric elliptic form operator is self-adjoint with compact resolvent
- Adjoint of a densely defined operator
- The adjoint is well defined, closed, and reverses inclusions
- Self-adjoint nonpositive operators generate bounded analytic semigroups
- Sectorial operator with the semigroup sign convention
- Complex sector and bounded analytic semigroup
- Semigroup sign and generator conventions
- Quadratic spectral bounds control a self-adjoint parabolic semigroup
- The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction
- The Poincare constant is the reciprocal square root of the first Dirichlet eigenvalue
- The Rayleigh principle for the first Dirichlet eigenvalue
- Discrete spectrum of a symmetric elliptic Dirichlet operator
- The Axiom of Countable Choice (AC_ω)
- The sharp Dirichlet Poincare inequality on an interval
- Global H² Dirichlet regularity
- Higher-order boundary regularity for Dirichlet problems
- Smoothing estimates for the semigroup generated by a sectorial operator
- The generator of the contour semigroup is the sectorial operator
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Divergence on a bounded C1 Euclidean domain
- Divergence for finite piecewise C1 presentations
- Classical normal derivative
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Harmonic functions are real analytic
- Choice function
- Complex norm and complex normed space
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complete metric space: every Cauchy sequence converges in the space
- Realification preserves a complex norm and its metric
- Strongly measurable Banach-valued function
- The Banach-valued simple integral is well defined
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Finite, countably infinite, countable, uncountable
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The space Lᵖ(μ) as the quotient by null functions
- The essential supremum of a measurable function with respect to a measure
- The class L¹(μ) of integrable functions
- Real and imaginary parts, complex conjugation, and modulus
- The spaces C_c(ℝⁿ) and C_c^∞(ℝⁿ)
- The space C₀(ℝⁿ) of continuous functions vanishing at infinity
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Threshold characterisations of real-valued and extended-real-valued measurability
- The Dirac set function at a point
- A Dirac set function is a probability measure
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- Integrable real and complex functions, and their integrals
- Continuity of a map of topological spaces at a point and globally
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- Separated sets, disconnection, and connected subset of ℝ
- Test function topology
- Test function operations are continuous
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Order on the reals
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Complete ordered field (least-upper-bound property)
- Ordered field
- Weak derivative of a locally integrable function
- Weak differentiation ignores null-set changes
- Uniqueness of a weak derivative as an almost-everywhere class
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- Complex Holder, Minkowski, and the quotient norm
- The dual space X^* of a normed space and its dual norm
- The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology
- Classical derivatives agree with weak derivatives
- The Sobolev norm descends to equivalence classes
- Linearity, locality, and commutation of weak derivatives
- Integral over a measurable subset
- Compactly supported scaled Euclidean bumps
- The standard smooth step function
- Explicit compactly supported smooth cutoffs
- Locally integrable functions as regular distributions
- Leibniz rule for distributions
- Multiplication of a distribution by a smooth function
- Sobolev extension domains and extension operators
- Integer-order Sobolev extension from a half-space
- Cᵏ boundary flattening preserves local W^k,p
- Bounded restriction and cutoff localisation in Sobolev spaces
- Monotone convergence for the integral
- Weighted tempered-distribution characterization of Hˢ
- Real-order Bessel-potential completion Hˢ
- The Bessel completion embeds canonically in tempered distributions
- Real powers of the Japanese bracket act on Schwartz space
- Smooth polynomially bounded multipliers on schwartz space
- Regular distribution from a locally integrable function
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Corollary
The Dirichlet Laplacian generates an analytic heat semigroup
Corollary
First Green identity
Corollary
Unique continuation for harmonic functions
Definition
The Axiom of Choice
Definition
Banach space
Definition
Bochner-integrable function
Definition
Bounded Cᵏ domains and boundary charts
Definition
Bounded C1 domains and their outward normals
Definition
A bounded linear operator between normed spaces
Definition
Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
Definition
Complex Lp classes and Euclidean test-function conventions
Definition
Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
Definition
Distribution
Definition
Distributional derivative
Definition
Interior, closure, boundary, exterior, derived set and isolated point in a topological space
Definition
Open ball, closed ball and sphere in a metric space
Definition
Integer-order Sobolev spaces and their norms
Definition
Weak topology on a normed space
Definition
Zero-boundary Sobolev space as a norm closure
Lemma
Compactly supported Sobolev functions extend by zero in every integer order
Lemma
Finite ambient partitions near compact sets
Lemma
Test function cutoffs and euclidean localization
Lemma
Weak Leibniz rule with a smooth factor
Theorem
Bounded Cᵏ domains admit integer-order Sobolev extension
Theorem