In Banach space theory (closed) subspaces and quotient seem to play a symmetric role. However, since the behavior of subspaces is more intuitive, subspaces appear more frequently. E.g., the theory of Schauder basis.

In the case of operators between reflexive Banach spaces, the symmetry is perfect. Passing to dual spaces and conjugate operators, we get the quotient version of a subspace concept or result, and vice versa. Although this is not so clear in the non-reflexive case, many concepts and results formulated for subspaces admit a corresponding version for quotients:

Injective spaces (extension property) and surjective spaces (lifting property);

Strictly singular and strictly cosingular operators;

Subspace incomparability (no inf. dim. subspace of X is isomorphic to a subspace of Y) and quotient incomparability;


Q$_1$ Is there any general principle implying that the role of subspaces and quotients is non-symmetric?

Q$_2$ Is there any concrete subspace (quotient) result or concept that does not admit a corresponding quotient (subspace) version?

Q$_3$ Is there any concrete topic in which the theory for subspaces is intrinsically richer than the corresponding theory for quotients?

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    $\begingroup$ Very interesting question. I wonder if there is a tag that could be added to bring it to the attention of researchers (not in Banach space theory) who may have studied analogous problems in other fields? $\endgroup$ – Philip Brooker Jun 12 '15 at 12:13
  • $\begingroup$ This question brings to mind the open problem of whether every infinite dimensional Banach space admits an infinite dimensional separable quotient. This problem seems to be hard, in stark contrast to the trivial problem of finding a separable infinite dimensional subspace. $\endgroup$ – Philip Brooker Jun 13 '15 at 1:44
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    $\begingroup$ A concrete answer to Q$_3$: every surjective space is isomorphic to $\ell_1(\Gamma)$, which is a dual space. However there are injective spaces not isomorphic to $\ell_\infty(\Gamma)$; even not isomorphic to a dual space. See Corollary 4.4 in [Rosenthal; Acta Math. 124 (1970), 205-248]. $\endgroup$ – M.González Jun 13 '15 at 9:16
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    $\begingroup$ For Q2: Let $X$ be reflexive. If $X$ embeds into $Y^*$, then $X^*$ is a quotient of $Y$. If $X$ is a quotient of $Y^*$, then $X^*$ need not embed into $Y$. $$$$ Another example: A quotient of a WCG space is WCG, but a subspace of a WCG space need not be WCG. (WCG = weakly compactly generated.) $\endgroup$ – Bill Johnson Jun 13 '15 at 20:29

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