Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 145-170.

THE QUOTIENT SHAPES OF l_p AND L_p SPACES

Nikica Uglešić

Abstract.   All l_p spaces (over the same field), p ≠ ∞, have the finite quotient shape type of the Hilbert space l₂. It is also the finite quotient shape type of all the subspaces l_p(p'), p < p' ≤ ∞, as well as of all their direct sum subspaces F₀^N(p'), 1 ≤ p' ≤ ∞. Furthermore, their countable and finite quotient shape types coincide. Similarly, for a given positive integer, all L_p spaces (over the same field) have the finite quotient shape type of the Hilbert space L₂, and their countable and finite quotient shape types coincide. Quite analogous facts hold true for the (special type of) Sobolev spaces (of all appropriate real functions).

2010 Mathematics Subject Classification.   54E99, 55P55.

Key words and phrases.   Normed (Banach, Hilbert vectorial) space, quotient normed space, l_p space, L_p space, Sobolev space, (algebraic) dimension, (infinite) cardinal, (general) continuum hypothesis, quotient shape.

DOI: https://doi.org/10.21857/9xn31cr62y

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