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)?

My apologies in advance if the question has been already asked.

**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).

non-linearmaps? $\endgroup$ – Yemon Choi Aug 18 '11 at 18:47