| E | separated semi-normed space |
| ǁ ǁE;ν | semi-norm of E of index ν |
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set indexing the semi-norms of E |
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equality of families of semi-norms |
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topological equality |
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topological equality up to an isomorphism |
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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 |
| ℒℓ(E1 × … × Eℓ; F) | space of continuous multilinear mappings |
POINTS AND SETS IN ℝd
| 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 |
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Ω1/n truncated by |
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part of Ω1/n which is star-shaped with respect to a, and its figure |
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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
| 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,…}
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