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Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. [note 3] For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system. An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This ...
Schur's lemma is frequently applied in the following particular case. Suppose that R is an algebra over a field k and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over k. If M is finite-dimensional, this division algebra is finite-dimensional.
In order to establish the Dolbeault isomorphism we need to prove the Dolbeault–Grothendieck lemma (or ¯-Poincaré lemma). First we prove a one-dimensional version of the ∂ ¯ {\displaystyle {\bar {\partial }}} -Poincaré lemma; we shall use the following generalised form of the Cauchy integral representation for smooth functions :
The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition (B). [5] (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lemma.
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space.
An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions.
For first-order logic, the theorem states that, given a theory T in the language L' ⊇ L and a formula φ in L', then the following are equivalent: for any two models A and B of T such that A|L = B|L (where A|L is the reduct of A to L), it is the case that A ⊨ φ[a] if and only if B ⊨ φ[a] (for all tuples a of A); φ is equivalent modulo ...
In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.