Math. Nachr. 290, No. 8–9, 1177–1186 (2017) / DOI 10.1002/mana.201600236 Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1] d Fernando Albiac ∗1 and Jos´ e Luis Ansorena ∗∗2 1 Mathematics Department, Public University of Navarre, Pamplona, 31006, Spain 2 Department of Mathematics and Computer Science, University of La Rioja, Logro˜ no, 26004, Spain Received 2 June 2016, revised 22 August 2016, accepted 4 September 2016 Published online 10 October 2016 Key words Isomorphic classification, Besov spaces, sequence spaces, quasi-Banach spaces MSC (2010) 46B03, 46B25, 42B35, 46B45, 46E35 The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1] d . Using the identification of the Besov space B α p,q ([0, 1] d ) with the ℓ q -infinite direct sum (⊕ ∞ n=1 ℓ n p ) q of finite-dimensional spaces ℓ n p (which holds independently of the dimension d ≥ 1 and of the smoothness degree of the space α> 0) we show that B α p,q ([0, 1] d ),0 < p, q ≤∞, is a family of mutually non-isomorphic spaces. The only exception is the isomorphism between the spaces B α 2,q ([0, 1] d ) and B α q,q ([0, 1] d ), which follows from Pelczy´ nski’s isomorphism between (⊕ ∞ n=1 ℓ n 2 ) q and ℓ q . We also tell apart the isomorphic classes of spaces B α p,q ([0, 1] d ) from the isomorphic classes of Besov spaces over the Euclidean space R d . C  2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction, background, and notation It is known that for p, q ∈ (0, ∞] and d ∈ N, Besov spaces over R d of indexes p and q are isomorphic to spaces ℓ q (ℓ p ), the direct ℓ q -sum of countably many copies of ℓ p , regardless of their degree of smoothness (see [17], [18], [28], [36]; cf. [2] for a recent detailed exposition on the evolution of this topic). In [37, footnote of page 242] Triebel stated that he had learnt from Pelczy´ nski that, for 1 < p < ∞ and 1 ≤ q ≤∞, different ℓ q (ℓ p ) spaces were not isomorphic. In the articles [10]–[12], Cembranos and Mendoza provided a proof to Pelczy´ nski’s claim and improved it by including the spaces ℓ ∞ (ℓ 1 ), c 0 (ℓ p ) and ℓ q (c 0 ) for 1 ≤ p, q ≤∞ in the list of mutually non-isomorphic Banach spaces. Finally, the authors completed in [1] the picture by letting the nonlocally convex relatives to be part of their natural family, i.e., by making the result extensive to the whole range of values 0 < p, q ≤∞. Truth be told, the spaces c 0 (ℓ p ) and ℓ q (c 0 ) for 0 < p, q < 1 were not explicitly included in the main theorem from [1], but the arguments in the proofs cover trivially these cases too. Thus Theorem 1.1 is the end of the story as far as the isomorphic classification of Besov spaces B α p,q (R d ) is concerned. Following the notation introduced by Bourgain et al. in [7], we write Z p,q = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ℓ q (ℓ p ) if 0 < p, q ≤∞; c 0 (ℓ p ) if 0 < p ≤∞, q = 0; ℓ q (c 0 ) if p = 0, 0 < q ≤∞; c 0 (c 0 ) if p = q = 0. Theorem 1.1 ([1], [10]–[12]) Let 0 ≤ p 1 , p 2 , q 1 , q 2 ≤∞. Then the spaces Z p 1 ,q 1 and Z p 2 ,q 2 are isomorphic if and only if p 1 = p 2 and q 1 = q 2 . Motivated by this positive result one may wonder about the corresponding isomorphic classification of Besov spaces in other contexts. Here we deal with Besov spaces defined on [0, 1] d , which are usually denoted by B α p,q ([0, 1] d ). ∗ Corresponding author: e-mail: fernando.albiac@unavarra.es, Phone: +34 948 169 553, Fax: +34 948 166 057 ∗∗ e-mail: joseluis.ansorena@unirioja.es, Phone: +34 941 299 464, Fax: +34 941 299 460 C  2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim