Frechet Calculus

$$ % ============================================ % GENERAL / UNIVERSAL % Used throughout the documentation % ============================================ % --- Basic Number Systems --- \newcommand{\IR}{\mathbb{R}} % Real numbers #listed \newcommand{\IC}{\mathbb{C}} % Complex numbers #listed \newcommand{\IN}{\mathbb{N}} % Natural numbers #listed \newcommand{\IZ}{\mathbb{Z}} % Integers #listed \newcommand{\IQ}{\mathbb{Q}} % Rational numbers #listed \newcommand{\IA}{\mathbb{A}} % Affine space #listed \newcommand{\IB}{\mathbb{B}} % Generic field #listed \newcommand{\ik}{\Bbbk} % Ground field #listed \newcommand{\ID}{\mathbb{D}} % Generic field #listed \newcommand{\IF}{\mathbb{F}} % Generic field #listed \newcommand{\IH}{\mathbb{H}} % Quaternions #listed \newcommand{\II}{\mathbb{I}} % Generic field #listed \newcommand{\IL}{\mathbb{L}} % Generic field #listed \newcommand{\IP}{\mathbb{P}} % Projective space #listed \newcommand{\IS}{\mathbb{S}} % Sphere #listed \newcommand{\IT}{\mathbb{T}} % Tensor algebra #listed \newcommand{\IV}{\mathbb{V}} % Generic vector space #listed % --- Function Spaces --- \newcommand{\CINF}{\mathcal{C}^\infty} % Smooth (infinitely differentiable) functions #listed \newcommand{\CC}{\mathcal{C}} % C^k functions #listed \newcommand{\Ck}{\mathcal{C}^k} % C^k functions #listed \newcommand{\CK}{\mathcal{C}^K} % C^k functions #listed % --- Fundamental Operators --- \newcommand{\del}{\partial} % Partial derivative #listed \newcommand{\uDelta}{\underline{\Delta}} % Discrete difference operator #listed \newcommand{\Shift}{\mathrm{S}_\downarrow} % Shift operator #listed % --- Logical & Set Operations --- \newcommand{\IFF}{\Leftrightarrow} % If and only if #listed \newcommand{\Ind}{\mathbb{1}} % Indicator/characteristic function #listed \newcommand{\IndA}[1]{\mathbb{1}\lbrace #1 \rbrace} % Indicator with condition #listed \newcommand{\1}{\mathbb{1}} % Indicator shorthand #listed \newcommand{\set}[1]{\{#1\}} % Simple set braces #listed \newcommand{\Set}[2]{\left\{\, #1 \;\vert\; #2 \,\right\}} % Set-builder notation #listed \newcommand{\CSet}[2]{\#\{\, #1 \;\vert\; #2 \,\right\}} % Cardinality notation #listed \newcommand{\C}{\,\#} % Cardinality operator #listed % --- Limits & Categorical --- \newcommand{\limproj}{\varprojlim} % Inverse limit #listed \newcommand{\limind}{\varinjlim} % Direct limit #listed \newcommand{\Hom}{\mathrm{Hom}} % Homomorphism #listed \newcommand{\End}{\mathrm{End}} % Endomorphism #listed \newcommand{\Ext}{\mathrm{Ext}} % Ext functor #listed % --- Arrows & Relations --- \newcommand{\ra}{\rightarrow} % Right arrow #listed \newcommand{\lra}{\longrightarrow} % Long right arrow #listed \newcommand{\xlra}[1]{\overset{#1}{\lra}} % Labeled long arrow #listed \newcommand{\la}{\leftarrow} % Left arrow #listed \newcommand{\lla}{\longleftarrow} % Long left arrow #listed \newcommand{\mono}{\hookrightarrow} % Monomorphism #listed \newcommand{\epi}{\twoheadrightarrow} % Epimorphism #listed \newcommand{\isom}{\cong} % Isomorphism #listed \newcommand{\downto}{\searrow} % Diagonal arrow #listed % --- Tensor & Algebraic Structures --- \newcommand{\tensor}{\otimes} % Tensor product #listed \newcommand{\tensors}{\tensor\dots\tensor} % Multiple tensor products #listed \newcommand{\Tensor}{\bigotimes} % Big tensor product #listed \newcommand{\stensor}{\odot} % Symmetric tensor product #listed \newcommand{\vsum}{\oplus} % Direct sum #listed \newcommand{\Vsum}{\bigoplus} % Big direct sum #listed % --- Calligraphic Letters (Generic) --- \newcommand{\KA}{\mathcal{A}} \newcommand{\KB}{\mathcal{B}} \newcommand{\KC}{\mathcal{C}} \newcommand{\KD}{\mathcal{D}} \newcommand{\KF}{\mathcal{F}} \newcommand{\KH}{\mathcal{H}} \newcommand{\KI}{\mathcal{I}} \newcommand{\KL}{\mathcal{L}} \newcommand{\KN}{\mathcal{N}} \newcommand{\KP}{\mathcal{P}} \newcommand{\KQ}{\mathcal{Q}} \newcommand{\KR}{\mathcal{R}} \newcommand{\KS}{\mathcal{S}} \newcommand{\KV}{\mathcal{V}} \newcommand{\KZ}{\mathcal{Z}} % --- Fraktur Letters (Generic) --- \newcommand{\gc}{\mathfrak{C}} \newcommand{\gd}{\mathfrak{D}} \newcommand{\gM}{\mathfrak{M}} \newcommand{\gm}{\mathfrak{m}} \newcommand{\gf}{\mathfrak{f}} \newcommand{\gu}{\mathfrak{U}} \newcommand{\fa}{\mathfrak{a}} \newcommand{\fg}{\mathfrak{g}} \newcommand{\fn}{\mathfrak{n}} \newcommand{\fk}{\mathfrak{k}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\fp}{\mathfrak{p}} \newcommand{\fP}{\mathfrak{P}} % \fg already defined above (line 95) as \mathfrak{g} % --- Text & Formatting --- \newcommand{\qtext}[1]{\quad\text{#1}\quad} % Quad spaced text #listed \newcommand{\stext}[1]{\;\text{#1}\;} % Small spaced text #listed \newcommand{\ssum}[1]{\sum_{\substack{#1}}} % Sum with substacked condition #listed \newcommand{\half}{\frac{1}{2}} % One-half #listed \newcommand{\floor}[1]{\lfloor #1 \rfloor} % Floor function #listed \newcommand{\ceil}[1]{\lceil #1 \rceil} % Ceiling function #listed \newcommand{\nl}{\\} % Newline #listed % --- Common Functions --- \newcommand{\id}{\mathrm{id}} % Identity function #listed \newcommand{\len}{\mathrm{len}} % Length of a word #listed \newcommand{\rk}{\mathrm{rk}} % Rank #listed \newcommand{\diag}{\operatorname{diag}} % Diagonal matrix #listed \newcommand{\Res}{\operatorname{Res}} % Residue #listed \newcommand{\Ker}{\mathrm{Ker}} % Kernel #listed \newcommand{\Diff}{\mathrm{Diff}} % Diffeomorphism group #listed \newcommand{\Pic}{\mathrm{Pic}} % Picard group #listed \newcommand{\Spec}{\mathrm{Spec}} % Spectrum #listed \newcommand{\D}{\mathrm{D}} % Differential operator #listed \newcommand{\DP}{\mathrm{D_{\!+}}} % Positive differential #listed \newcommand{\DDP}{\mathrm{D^{\!+}}} % Upper differential #listed % --- Variants --- \newcommand{\vphi}{\varphi} % Variant phi #listed \newcommand{\sphi}{\phi} % Straight phi #listed \newcommand{\eps}{\varepsilon} % Epsilon variant #listed \newcommand{\pt}{*} % Point notation #listed \newcommand{\point}{*} % Point notation alt #listed % --- Set Operations --- \newcommand{\union}{\cup} % Union #listed \newcommand{\Union}{\bigcup} % Big union #listed \newcommand{\dotcup}{\ensuremath{\mathaccent\cdot\cup}} % Disjoint union #listed \newcommand{\dunion}{\dotcup} % Disjoint union alt #listed % \< and \> removed - conflict with LaTeX tabbing commands and unused in docs \newcommand{\inpart}[1]{\in\text{\part}(#1)} % In partition of #listed \newcommand{\trl}{\triangleleft} % Left triangle #listed \newcommand{\trr}{\triangleright} % Right triangle #listed % --- Misc --- \newcommand{\curly}[1]{\mathcal{#1}} \newcommand{\op}[1]{\mathrm{#1}} \newcommand{\Cat}[1]{\mathfrak{#1}} \newcommand{\cat}[1]{\mathbf{#1}} \newcommand{\CAT}[1]{\left\{\,\text{#1}\,\right\}} \newcommand{\CATii}[2]{\left\{\,\begin{array}{c}\text{#1}\\\text{#2}\end{array}\,\right\}} \newcommand{\Mon}{\mathrm{Mon}} \newcommand{\Lin}{\mathrm{Lin}} \newcommand{\ev}{\mathrm{ev}} \newcommand{\tc}{\prec_{\mathrm{tc}}} \newcommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \newcommand{\QED}{\square} % End of proof #listed \renewcommand{\part}{\vdash} % Turnstile #listed (overrides LaTeX \part sectioning) \newcommand{\opart}{\models} % Semantic entailment #listed \newcommand{\Def}{\mathrm{Def}} % Bialgebra defect #listed % ============================================ % CHAPTER 1: DIFFERENTIAL DUALITY % ============================================ \newcommand{\Jet}{\mathbf{Jet}} % Jet space of C infinity germs #listed \newcommand{\jet}{\mathrm{jet}} % Jet functor #listed \newcommand{\E}{\mathbf{E}} % Space of smooth functions #listed \newcommand{\EE}{\mathbf{E}} % Space of smooth functions alt #listed % ============================================ % CHAPTER 2: DISCRETE DUALITY % ============================================ % --- Multisets and Iterated Structures --- \newcommand{\KM}{\mathcal{M}} % Multiset / multi-index set #listed \newcommand{\ualpha}{\underline{\alpha}} % Underline alpha (iterated multi-index) #listed \newcommand{\ubeta}{\underline{\beta}} % Underline beta (iterated multi-index) #listed \newcommand{\ugamma}{\underline{\gamma}} % Underline gamma (iterated multi-index) #listed \newcommand{\uS}{\underline{S}} % Underline S (iterated subset) #listed \newcommand{\uT}{\underline{T}} % Underline T (iterated subset) #listed % --- From Affine Cubes --- \newcommand{\BC}{\mathbf{B}} % Affine Cube space #listed % --- From Cubes.md and affine cube space --- \newcommand{\BB}{\mathbb{B}} % Boolean lattice #listed \newcommand{\minelt}{\hat{0}} % Minimal element #listed \newcommand{\maxelt}{\hat{1}} % Maximal element #listed \newcommand{\sleq}{\subseteq} % Subset relation #listed \newcommand{\drk}{\mathrm{rk_{\Delta}}} % Discrete rank #listed \newcommand{\Fun}{\operatorname{F}} % Function space #listed \newcommand{\Cube}{\operatorname{Cube}} % Geometric cubes #listed \newcommand{\Cu}{\mathrm{Cube}} % Cube spaces (NC calculus); symbol choice revisable #listed \newcommand{\Fl}{\operatorname{Fl}} % Flow (NC calculus) #listed \newcommand{\Aff}{\operatorname{Aff}} % Affine cubes #listed \newcommand{\Exp}{\operatorname{exp}} % Geometric realization (zeta transform) #listed \newcommand{\Log}{\operatorname{log}} % Tangent logarithm (Möbius transform) #listed \newcommand{\aprod}{\star} % Anchored product of cubes #listed \newcommand{\acoprod}{\Delta} % Anchored coproduct of cubes #listed \newcommand{\tcoprod}{\Delta^\tau} % Transport coproduct of cubes #listed \newcommand{\tprod}{\star^\tau} % Transport product of cubes #listed \newcommand{\Part}{\mathrm{Part}} % Partitions of a set #listed \newcommand{\Cov}{\mathrm{Cov}} % Coverings #listed \DeclareMathOperator{\lf}{\mathrm{leaf}} % Leaf support #listed \newcommand{\CT}{\mathbf{CT}} % Covering trees #listed \newcommand{\BH}{\mathbf{IH}} % Iterated hypergraphs #listed \newcommand{\IHCov}{\mathbf{IHCov}} % Iterated hypergraph covers #listed \newcommand{\vac}{{|0\rangle}} % Empty partition #listed % --- From Filtered Vector Spaces.md --- \newcommand{\gr}{\mathrm{gr}} % Graded/associated graded object #listed % ============================================ % CHAPTER 3: ULTRA CALCULUS % ============================================ \newcommand{\POS}{\mathbf{Pos}} % Growth domain spaces #listed \newcommand{\U}{\mathbf{U}} % Ultra regulator quotient #listed \newcommand{\IU}{\mathbb{U}} % Ultra regulator #listed \newcommand{\IG}{\mathbb{G}} % Growth profile #listed \newcommand{\Tame}{\mathbf{Tame}} % Tame growth functions #listed \newcommand{\Scale}{\mathrm{Scale}} % Scaling operator #listed \newcommand{\Bell}{\mathcal{B}} % Bell polynomials #listed \newcommand{\uexp}{\exp_+} % Exponential generating series #listed % --- Ultra Structures --- \newcommand{\Sym}{\mathbf{S}} % Symmetric functions/algebra #listed \newcommand{\SYM}{\mathrm{Sym}} % Symmetric algebra, roman #listed \renewcommand{\SS}{\Sym} % Symmetric functions alt #listed (overrides LaTeX \SS) \newcommand{\SP}{\Sym^+} % Positive symmetric functions #listed \newcommand{\SH}{\hat{\Sym}} % Completed symmetric functions #listed \newcommand{\SPH}{\hat{\Sym}^+} % Completed positive symmetric #listed % --- Completed tangent/cotangent spaces (DDC paper) --- \newcommand{\STH}{\hat{S}T} % Completed symmetric cotangent space #listed \newcommand{\CTH}{\hat{C}T} % Completed co-cube space #listed % --- Ultra Operators --- \newcommand{\MIX}{\mathrm{Mix}} % Mixing operator #listed \newcommand{\BMIX}{\mathrm{BMix}} % Boolean mixing #listed \newcommand{\TMIX}{\mathrm{TMix}} % Transport mixing #listed \newcommand{\TBMIX}{\mathrm{TMix}} % Transport boolean mixing #listed \newcommand{\TRANS}{\mathrm{Trans}} % Transport operator #listed % --- Gauge & Equivalence --- \newcommand{\gaugeleq}{\preccurlyeq} % Gauge less-or-equal #listed \newcommand{\gaugeeq}{\asymp} % Gauge equivalence #listed \newcommand{\gaugegeq}{\preccurlygeq} % Gauge greater-or-equal #listed \newcommand{\tstar}{\circledast} % Tight star product #listed % --- Forward Differences --- \newcommand{\FD}{\blacktriangle} % Forward difference #listed \newcommand{\fd}{\FD} % Forward difference alt #listed \newcommand{\AC}{\square} % Associated character #listed % ============================================ % OPERATORS (DeclareMathOperator) % ============================================ \DeclareMathOperator{\Supp}{\mathrm{Supp}} % Support #listed \DeclareMathOperator{\supp}{\mathrm{Supp}} % Support #listed \newcommand{\esssupp}{\operatorname*{ess-supp}} % Support #listed \DeclareMathOperator{\Alt}{\Lambda} % Alternating/exterior #listed \DeclareMathOperator{\ad}{ad} % Adjoint representation #listed \DeclareMathOperator{\ch}{ch} % Chern character #listed \DeclareMathOperator{\td}{td} % Todd class #listed \DeclareMathOperator{\TD}{TD} % Todd operator #listed \DeclareMathOperator{\pr}{pr} % Projection #listed \DeclareMathOperator{\Map}{Map} % Mapping space #listed \DeclareMathOperator{\Pol}{Pol} % Polarization % ============================================ % ENVIRONMENT SIDEBARS % Plain neutral-grey left rule on theorem-like environments, % marking structure without web-style color coding. % ============================================ \usepackage{xcolor} \usepackage[framemethod=TikZ]{mdframed} \usepackage{booktabs} % pandoc emits \toprule/\midrule/\bottomrule for pipe tables \definecolor{envRule}{HTML}{666666} \mdfdefinestyle{envstyle}{% linewidth=0.8pt, topline=false, bottomline=false, rightline=false, leftline=true, leftmargin=-8pt, rightmargin=0pt, innerleftmargin=7.2pt, innerrightmargin=0pt, innertopmargin=0pt, innerbottommargin=0pt, skipabove=1em, skipbelow=1em, linecolor=envRule, } \surroundwithmdframed[style=envstyle]{theorem} \surroundwithmdframed[style=envstyle]{proposition} \surroundwithmdframed[style=envstyle]{lemma} \surroundwithmdframed[style=envstyle]{corollary} \surroundwithmdframed[style=envstyle]{definition} \surroundwithmdframed[style=envstyle]{example} \surroundwithmdframed[style=envstyle]{remark} \surroundwithmdframed[style=envstyle]{proof} $$