Continuous Functions. Jacques Simon

E	separated semi-normed space
ǁ ǁE;ν	semi-norm of E of index ν
	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 × … × Eℓ	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 × … × Eℓ; F)	space of continuous multilinear mappings

      POINTS AND SETS IN ℝ^d

ℝ^d	Euclidean space: ℝ^d = {x = (x1, …, xd) : ∀i, xi ∈ ℝ}
|x|	Euclidean norm:
x · y	Euclidean scalar product: x · y = x1y1 + … + xdyd
ei	ith basis vector of ℝ^d
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 ℝ^d
|ω|	Lebesgue measure of the open set ω
σ	negligible subset of ℝ^d
B(x, r)	closed ball B(x, r) = {y ∈ ℝ^d : |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 ∈ ℝ^d : |y − x| = r}
Δs,n	closed cube of edge length 2^−n centered at 2^−ns
P(υ^1,…, υd)	open parallelepiped with edges υ^1, …, υd

      OTHER SETS

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,…}