.. File: appendix_symbols.rst .. Author: William DeMeo .. Date: 2019.10.24 .. Copyright (c) 2019 William DeMeo (see the LICENSE file) .. include:: _static/math_macros.rst .. _appendix-symbols: ======================= symbols ======================= .. contents:: :local: :depth: 1 .. _acronyms-and-abreviations: Acronyms and abbreviations --------------------------- .. glossary:: AC :term:`absolutely continuous ` a.e. almost everywhere cod codomain dcpo :term:`directed-cocomplete poset` dct :term:`dominated convergence theorem` mct :term:`monotone convergence theorem` dom domain em :term:`law of the excluded middle` ω-cpo :term:`ω-chain cocomplete poset` ran range ---------------------------------------------------- .. _symbol-commands: Symbols ------- The list below shows what to type (e.g., in the vscode IDE with lean extension) to produce some special characters. .. glossary:: Ā ``A\bar`` \ (the closure of the set A) B̄ ``B\bar`` \ (the closure of the set B) 𝔸 ``\BbbA`` 𝔹 ``\BbbB`` ℂ ``\BbbC`` \ (complex numbers) ℕ ``\BbbN`` \ (natural numbers) ℝ ``\BbbR`` \ (real numbers) ℝ* ``\BbbR*`` or ``\BbbR^\ast`` \ (extended real numbers, :math:`ℝ^∗ = ℝ ∪ \{-∞, ∞\}`) 𝕋 ``\BbbT`` ℤ ``\BbbZ`` \ (integers) ℬ ``\mscrB`` \ (:math:`ℬ(X) =` :term:`Borel σ-algebra` of :math:`X`) L ``\mscrL`` 𝒫 ``\mscrP`` \ (𝒫(X) = the power set of X) 𝔅 ``\mfrakB`` \ (:math:`𝔅(X,Y) =` :term:`bounded linear transformations ` from :math:`X` to :math:`Y`) 𝔐 ``\mfrakM`` \ (usually a :term:`σ-algebra`) α ``\alpha`` \ (often used to denote an uncountable index) β ``\beta`` γ ``\gamma`` Γ ``\Gamma`` δ ``\delta`` \ (usually a small real number) Δ ``\Delta`` ε ``\epsilon`` \ (usually a small real number) ι ``\iota`` κ ``\kappa`` \ (usually a large cardinal number) λ ``\lambda`` \ (usually a measure) Λ ``\Lambda`` ρ ``\rho`` σ ``\sigma`` Σ ``\Sigma`` ∑ ``\sum`` ∏ ``\prod`` Π ``\Pi`` π ``\pi`` φ ``\varphi`` ϕ ``\phi`` Φ ``\Phi`` χ ``\chi`` \ (χₐ = the characteristic function of a) Χ ``\Chi`` ψ ``\psi`` Ψ ``\Psi`` ω ``\omega`` Ω ``\Omega`` ℵ ``\aleph`` 𝚤 ``\imath`` 𝚥 ``\jmath`` ℓ ``\ell`` fₗ ``f\_l`` f̄ ``f\bar`` f̂ ``f\hat`` f̃ ``f\tilde`` h₁ ``h\_1`` h₂ ``h\_2``, etc. ∩ ``\cap`` \ (intersection) ⋂ ``\bigcap`` \ (intersection) ∪ ``\cup`` \ (union) ⋃ ``\bigcup`` \ (union) ∧ ``\and`` or ``\wedge`` \ (infimum) ⋀ ``\bigwedge`` \ (infimum) ∨ ``\vee`` or ``\or`` \ (supremum) ⋁ ``\bigvee`` \ (supremum) ¬ ``\neg`` \ (negation) ∘ ``\o`` or ``\circ`` × ``\x`` or ``\times`` ∃ ``\exists`` ∀ ``\all`` or ``\forall`` ∈ ``\in`` ∋ ``\ni`` ∉ ``\notin`` ∌ ``\nni`` ≤ ``\leq`` ≥ ``\geq`` ⊆ ``\subseteq`` ⊇ ``\supseteq`` ⊂ ``\subset`` ⊃ ``\supset`` ≪ ``\ll`` \ (:math:`λ ≪ μ` means ":math:`λ` is :term:`absolutely continuous ` with respect to :math:`μ`") ≫ ``\gg`` ⋆ ``\star`` ∗ ``\ast`` ≈ ``\approx`` ∼ ``\sim`` ≡ ``\equiv`` ≅ ``\cong`` \ (isomorphism or congruence relation) ⟨ ``\langle`` ⟩ ``\rangle`` ← ``\leftarrow`` → ``\to`` or ``\rightarrow`` ↑ ``\uparrow`` ↓ ``\downarrow`` ⟹ ``\Longrightarrow`` ⟺ ``\iff`` ↦ ``\mapsto`` ↠ ``\twoheadrightarrow`` (a surjective map) ↣ ``\rightarrowtail`` (an injective map) ⤖ ``\twoheadrightarrowtail`` (a bijective map) ∅ ``\emptyset`` ⊢ ``\vdash`` ⊨ ``\vDash`` ⫢ ``\vDdash`` ⊧ ``\models`` ⋈ ``\bowtie`` ---------------------------------