However, in spherical polar coordinates the “ r direction” is surely not the same everywhere and we need to define it unambiguously. That is simple and straightforward because the “ x direction” is everywhere the same direction. In Cartesian coordinates, a unit vector e ˆ x is of unit length and in the x direction. Harris, in Mathematics for Physical Science and Engineering, 2014 Vectors in Curvilinear Coordinates All Banach spaces are presumed to be separable unless otherwise stated.įrank E. We begin with some results on the global structure of L p and in particular those involving the Haar basis.
L p-spaces up to isomorphism and this is discussed in Section 5 below. L ∞) spaces are given in, Much work was done to study and attempt to classify the It is known that if X contains an isomorph of L 1 then X is isomorphic to L 1 and if X embeds into ℓ 1 then X is isomorphic to ℓ 1. It is conjectured that every infinite dimensional complemented subspace X of L 1 is isomorphic to L 1 or ℓ 1. The situation for L 1 is more complicated. L p,λ for some λ and 1 < p < ∞ iff X is isomorphic to a complemented subspace of L p which is not isomorphic to Hilbert space. It ultimately turns out (see Section 5) that a separable X is L p,λ-space if for all finite dimensional spaces F ⊆ X there exists a finite dimensional E withĭ ( E, ℓ p dim E ) ≤ λ.
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