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Rather, it is isomorphic to a subgroup of Sp(2n, C), and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Sp(n) is the compact real form of the complex symplectic Lie algebra sp(2n, C). Sp(n) is a real Lie group with (real) dimension n(2n + 1). It is compact and simply ...
Define the bijection g(t) from T to (0, 1): If t is the n th string in sequence s, let g(t) be the n th number in sequence r ; otherwise, g(t) = 0.t 2. To construct a bijection from T to R, start with the tangent function tan(x), which is a bijection from (−π/2, π/2) to R (see the figure shown on the right).
Cantor's paradox is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important to note what this contradiction is.
Property (c) says the operator M is bounded on L p (R n); it is clearly true when p = ∞, since we cannot take an average of a bounded function and obtain a value larger than the largest value of the function. Property (c) for all other values of p can then be deduced from these two facts by an interpolation argument.
While there are several proofs of this theorem, a common one is given below: For p = ∞, the inequality is trivial (since the average of a function is no larger than its essential supremum). For 1≤ p < ∞, first we shall use the following version of the Vitali covering lemma to prove the weak-type estimate.
The Fourier transform is a linear isomorphism F:𝒮(R n) → 𝒮(R n). If f ∈ 𝒮(R n) then f is Lipschitz continuous and hence uniformly continuous on R n. 𝒮(R n) is a distinguished locally convex Fréchet Schwartz TVS over the complex numbers. Both 𝒮(R n) and its strong dual space are also: complete Hausdorff locally convex spaces ...
Computable ordinals (or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a computable function.There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set ...
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.