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This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and how they go about looking up information on such spaces, given that it does not seem easy to get focused results when searching online.

To be a little more precise: are such spaces referred to by anyone as "approximate $L_p$-spaces" or "local $L_p$-spaces"? It seems less than ideal to have an important notion described only by literal typography.

By the way: I am aware of the original papers of Lindenstrauss–Pelczynski and Lindenstrauss–Rosenthal, so that is not my question. My question is about the terminology or description that specialists in Banach space theory would use to refer to these spaces, when asking each other questions or giving each other outlines of proofs.

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    $\begingroup$ I think I've heard "script $L_p$-spaces". $\endgroup$
    – Mateusz Wasilewski
    Commented Jul 1, 2019 at 17:11
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    $\begingroup$ Yes, "script $L_p$ spaces" is what we say when talking. I do not know how to search for them online. The notation and name came long before the internet, so the inventors of the term did not consider the problem of searching. $\endgroup$
    – Bill Johnson
    Commented Jul 1, 2019 at 17:36
  • $\begingroup$ Thanks Mateusz and Bill. @BillJohnson: am I right that the Handbook does not have a particular article on the "script $L_p$ spaces"? $\endgroup$
    – Yemon Choi
    Commented Jul 1, 2019 at 18:12
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    $\begingroup$ @Yemon Choi: The Alspach-Odell article in the Handbook covers the script-$L_p$ spaces. $\endgroup$
    – Bill Johnson
    Commented Jul 1, 2019 at 19:59

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Just so that this question can be marked as answered: Bill Johnson states in comments that those in the know call these Banach spaces "script $L_p$-spaces", and I trust his awareness/judgement of the norms in the community.

I was hoping that in the years since the foundational work, other terminology might have emerged; but so it goes.

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    $\begingroup$ Exactly, that is why I always write a $\mathscr L_p$-space but an $L_p$-space. $\endgroup$
    – Tomasz Kania
    Commented Jul 2, 2019 at 3:46
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    $\begingroup$ "Local $L_p$" would be a better name because in Banach space theory a local property of a Banach space is a property that depends on the structure of the finite dimensional subspaces; so, for example, $\ell_p$ and $L_p$ have the same local structure. Two Banach spaces are the same locally if they have isomorphic ultra powers. $\endgroup$
    – Bill Johnson
    Commented Jul 2, 2019 at 16:05

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