منابع مشابه
A NOTE ON THE SPACES Lp FOR 0 < p < 1
It is shown that there is no Hausdorff vector topology p on the space Lp (where 0 < p < 1) such that the unit ball of Lp is relatively compact for the topology p. It is well known that the space L1 (0, 1) is not a dual Banach space; this follows from the Krein-Milman theorem. It is not even isomorphic to a dual space, by a result due to Gelfand [2] (see Bessaga and Pelczyn'ski [1] and Namioka [...
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For 0 < p < 1 we give examples of Banach spaces isometrically embedding into Lp but not into any Lr with p < r < 1.
متن کاملL p-SPACES FOR 0 < p < 1
In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...
متن کاملLp-Regularized Least Squares (0<p<1) and Critical Path
The least squares problem is formulated in terms of lp quasi-norm regularization (0 < p < 1). Two formulations are considered: (i) an lp-constrained optimization and (ii) an lp-penalized (unconstrained) optimization. Due to the nonconvexity of the lp quasi-norm, the solution paths of the regularized least squares problem are not ensured to be continuous. A critical path, which is a maximal cont...
متن کاملAffine Synthesis onto Lp when 0 < p ≤ 1
The affine synthesis operator Sc = ∑j>0 ∑k∈Zd cj,kψj,k is shown to map the coefficient space (Z+ × Zd ) surjectively onto Lp(Rd ), for p ∈ (0, 1]. Here ψj,k(x) = | det aj |ψ(aj x − k) for dilation matrices aj that expand, and the synthesizer ψ ∈ Lp(Rd ) need satisfy only mild restrictions, for example, ψ ∈ L1(Rd ) with nonzero integral or else with periodization that is real-valued, nontrivial ...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1940
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1940-07308-2