Continuous Functions. Jacques Simon

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set indexing the semi-norms of E equality of families of semi-norms topological equality topological equality up to an isomorphism topological inclusion E-weak space E endowed with pointwise convergence in Eʹ Eʹ dual of E Ed Euclidean product E × … × E E1 × … × E_ℓ product of spaces Ê sequential completion of E Ů interior of the set U Ū closure of U ∂U boundary of U [u, υ] closed segment: [u, υ] = {tu + (1 − t)υ : 0 ≤ t ≤ 1} ℒ(E; F) space of continuous linear mappings ℒ^ℓ(E₁ × … × E_ℓ; F) space of continuous multilinear mappings

      POINTS AND SETS IN ℝ^d

ℝd	Euclidean space: ℝd = {x = (x1, …, xd) : ∀i, xi ∈ ℝ}
|x|	Euclidean norm:
x · y	Euclidean scalar product: x · y = x1y1 + … + xdyd
ei	ith basis vector of ℝd
Ω	domain on which a function ƒ is defined
Ω D	domain of f ⋄ μ: ΩD = {x : x + D ⊂ Ω}, and its figure
Ω1/n	Ω with a neighborhood of the boundary of size 1/n removed
	Ω1/n truncated by
	part of Ω1/n which is star-shaped with respect to a, and its figure
	potato-shaped set:
κn	crown-shaped set:
ω	subset of ℝd
|ω|	Lebesgue measure of the open set ω
σ	negligible subset of ℝd
B(x, r)	closed ball B(x, r) = {y ∈ ℝd : |y − x| ≤ r}
Ḃ(x, r)	open ball Ḃ(x, r) = {y ∈ ℝd : |y − x| < r}
υd	measure of the unit ball: υd = |Ḃ(0, 1)|
C(x, p, r)	open crown C(x, p, r) = {y ∈ ℝd : ρ < |y − x| < r}
S(x,r)	sphere: S(x, r) = {y ∈ ℝd : |y − x| = r}
Δs,n	closed cube of edge length 2−n centered at 2−ns
P(υ1,…, υd)	open parallelepiped with edges υ1, …, υd

      OTHER SETS

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ℕ*	set of natural numbers: ℕ* = {0, 1, 2,…}
ℕ	set of non-zero natural numbers: ℕ = {1, 2,…}
ℤ	set of integers: ℤ = {…,−2, −1,0, 1, 2,…}
ℚ	set of rational numbers
ℝ	space of real numbers