## “Measuring” how far is one Banach space from being surjectively isometric to another

Bonjour/bonsoir à toutes et à tous.

Assume that $\mathbf{V} \equiv (V, \|\cdot\|_V)$ and $\mathbf{W} \equiv (W, \|\cdot\|_W)$ are Banach spaces (over the real or complex field).

Question 1. What are some appropriate indices you might use to "measure" how far is $\mathbf{V}$ from being i) surjectively isometric (see note N1) or ii) isometrically isomorphic to $\mathbf{W}$ (see note N2)?

I am conscious that the question may sound a little vague, so take the Banach-Mazur distance as a practical example of what I (am trying to) mean.

Added later. After an answer by Bill Johnson (see below), I'm adding here that another index (in the sense of Question 1) is given, for the non-linear case, by the Lipschitz distance (or Lipschitz distorsion). This is known to be the same as the Banach-Mazur distance so far as $\mathbf{V}$ and $\mathbf{W}$ are (isomorphic and) finite-dimensional. Yet, as still pointed out by BJ, the same question, when raised in the infinite-dimensional setting with regard to the separable case, is an open problem to date. A further possibility, when $\dim(V) = \dim(W) < \infty$, is given by the so-called weak Banach-Mazur distance (see my comment to Bill's first answer for a reference).

Question 2. Could you provide some concrete examples illustrating why, depending on the case, the one index should be preferred to the others (if any)?

Notes. (N1) Following a comment by Yemon Choi, I emphasize that, unless differently stated, I am using the term isometry to refer to both linear and non-linear isometries. (N2) Of course, in the real case, there is no true need to distinguish between conditions i) and ii) in the statement of Question 1 (by the Mazur-Ulam theorem).

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 I am slightly confused about terminology. In Question 1, are you interested in surjective, isometric, non-linear maps? – Yemon Choi Aug 18 2011 at 18:47 @Yemon Choi. Yes, I will edit the original post to make this definitely clear. – Salvo Tringali Aug 18 2011 at 19:44

For (i) the usual thing is to take the Lipschitz analogue of the Banach-Mazur distance; namely, the infimum over injective and surjective maps $T$ from $V$ to $W$ of the Lipschitz constant of $T$ times the Lipschitz constant of $T^{-1}$. Whether this is equivalent to the Banach-Mazur distance for separable Banach spaces is a well known open problem. See the book by Benyamini and Lindenstrauss.

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 @Bill Johnson. Thank you for your contribution and the reference. Just for the record, a further possibility, in the finite-dimensional case, is provided by the so-called weak Banach-Mazur distance as given in N. Tomczak-Jaegermann, "The weak distance between Banach spaces", Math. Nachr., 119 (1984), pp. 291-307. – Salvo Tringali Aug 18 2011 at 21:16 Well, sure, but the weak distance does not well measure farness from being isometric (nor does, e.g., the Gromov-Hausdorff distance). – Bill Johnson Aug 19 2011 at 0:00

Rather than talk about the weak distance and distance, it is better to discuss the weak factorization constant and the factorization constant of an operator $u$ through an operator $T$. The factorization constant of $u: X\to Y$ through $T:Z\to W$, $\gamma_T(u)$, is the infimum of $\|\alpha\|\cdot \|\beta\|$ over all $\alpha:X\to Z$ and $\beta:W\to Y$ for which $\beta T \alpha =u$. This measurement of the size of $u$ is generally not a norm, but you can convexify it to get the weak factorization constant, $\hat{\gamma}_T(u)$, of $u$ through $T$, which is defined to be the infimum of $\sum_i \gamma_T(u_i)$ s.t. $u=\sum_i u_i$. The (weak) factorization constant of $u$ through a space $Z$ is just the (weak) factorization constant of $u$ through $I_Z$. Obviously you can write down the distance and weak distance in terms of factorization and weak factorization constants.

One classical situation in which these parameters differ a lot is in my Studia Math. 89 (1988), 79--103 paper with Figiel and Schechtman. Let $u$ be the basis to basis mapping from $\ell_2^n$ to the first $n$ Rademacher functions in $L_1$. The factorization constant of this operator through $\ell_1^{Cn}$ is large for any fixed $C$, but the weak factorization constant through $\ell_1^n$ is bounded independently of $n$. That is, you cannot well factor this embedding of $\ell_2^n$ through a low dimensional $L_1$ space, but you can well weakly factor it through $\ell_1^n$ (in fact, any operator from $\ell_2^n$ into $L_1$ well weakly factors through $\ell_1^n$; see Proposition 5.5 of the paper I mentioned above). We also show that if you want to well factor this Rademacher embedding $u$ through $\ell_1^k$, then $k$ must be at least exponential in $n$.

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