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
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- The Axiom of Choice
- Spectral form domain and core of a semibounded operator
- Completion of a normed space
- The metric completion of a normed space carries a unique compatible Banach-space structure
- The norm completion of an inner-product space is a Hilbert space
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm
- Hilbert space
- Symmetric, self-adjoint and essentially self-adjoint operators
- Projection valued measure
- Banach space
- Compact linear operator
- Linear map between vector spaces over the same field
- The Axiom of Countable Choice (AC_ω)
- Strongly measurable Banach-valued function
- The space Lᵖ(μ) as the quotient by null functions
- Bochner Lp is complete for a Banach value space
- Realification preserves a complex norm and its metric
- Holder's inequality for integrals, including the endpoint cases
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Time translations and mollification in Bochner Lp
- Bochner-integrable function
- Bochner integrability criterion
- Bochner integral norm inequality
- Bounded linear maps commute with Bochner integration
- Test function space d of an open set
- Distributional derivative
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Fixed support test function frechet space
- Test function topology
- Linearity of the Bochner integral
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- Integer-order Sobolev spaces are Banach
- A closed subspace of a Banach space is Banach
- The dual space X^* of a normed space and its dual norm
- If Y is Banach then B(X,Y) is Banach
- Smooth compactly supported functions of an open set are dense in L²
- L² with the integral pairing is a Hilbert space
- Integer-order Sobolev extension from a half-space
- A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not
- A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
- A quotient of a Banach space by a closed subspace is Banach
- Topological vector spaces over the real and complex fields
- J. C. Bellido and G. García-Sáez, Bessel Potential Spaces and Complex Interpolation: Continuous embeddings, arXiv:2503.04310v6 · PDF pp.4–6, §2 before §2.1 (compatible couples, admissible operators, retractions); §2.2, PDF pp.9–10 (closed-strip class, Banach boundary norm, vector-valued strip maximum inequality and exact operator bound).
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
Lemma
Spectral cutoff cores and unitary identifications of completed scales
Definition
Compatible compact Banach triples
Lemma
Bochner spaces and bounded inclusion of value spaces
Definition
Distributional time derivatives with values in a Banach space
Lemma
The L2 pivot injects into the local negative Sobolev space
Lemma
One flat reflection with simultaneous Lq and W1q bounds
Lemma