"Measuring" how far is one Banach space from being surjectively isometric to another - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:24:10Z http://mathoverflow.net/feeds/question/73145 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73145/measuring-how-far-is-one-banach-space-from-being-surjectively-isometric-to-anot "Measuring" how far is one Banach space from being surjectively isometric to another Salvo Tringali 2011-08-18T12:56:24Z 2011-08-19T22:05:36Z <p>Bonjour/bonsoir à toutes et à tous.</p> <p>Assume that <code>$\mathbf{V} \equiv (V, \|\cdot\|_V)$</code> and <code>$\mathbf{W} \equiv (W, \|\cdot\|_W)$</code> are Banach spaces (over the real or complex field). </p> <blockquote> <p><strong>Question 1.</strong> What are some appropriate indices you might use to "measure" how far is <code>$\mathbf{V}$</code> from being i) surjectively isometric (see note N1) or ii) isometrically isomorphic to <code>$\mathbf{W}$</code> (see note N2)?</p> </blockquote> <p>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.</p> <p><strong>Added later.</strong> 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 $\dim(V) = \dim(W) &lt; \infty$, is given by the so-called weak Banach-Mazur distance (see my comment to Bill's first answer for a reference).</p> <blockquote> <p><strong>Question 2.</strong> Could you provide some concrete examples illustrating why, depending on the case, the one index should be preferred to the others (if any)?</p> </blockquote> <p>My apologies in advance if the question has been already asked.</p> <p><strong>Notes.</strong> (N1) Following a comment by Yemon Choi, I emphasize that, unless differently stated, I am using the term <em>isometry</em> to refer to both linear and non-linear isometries. (N2) Of course, in the real case, there is no <em>true</em> need to distinguish between conditions i) and ii) in the statement of Question 1 (by the Mazur-Ulam theorem).</p> http://mathoverflow.net/questions/73145/measuring-how-far-is-one-banach-space-from-being-surjectively-isometric-to-anot/73156#73156 Answer by Bill Johnson for "Measuring" how far is one Banach space from being surjectively isometric to another Bill Johnson 2011-08-18T14:31:11Z 2011-08-18T14:31:11Z <p>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. </p> http://mathoverflow.net/questions/73145/measuring-how-far-is-one-banach-space-from-being-surjectively-isometric-to-anot/73241#73241 Answer by Bill Johnson for "Measuring" how far is one Banach space from being surjectively isometric to another Bill Johnson 2011-08-19T20:52:41Z 2011-08-19T22:05:36Z <p>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 $T:Z\to W$, <code>$\gamma_T(u)$</code>, is the infimum of <code>$\|\alpha\|\cdot \|\beta\|$</code> over all <code>$\alpha:X\to Z$</code> and <code>$\beta:W\to Y$</code> for which <code>$\beta T \alpha =u$</code>. This measurement of the size of $u$ is generally not a norm, but you can convexify it to get the weak factorization constant, <code>$\hat{\gamma}_T(u)$</code>, of $u$ through $T$, which is defined to be the infimum of <code>$\sum_i \gamma_T(u_i)$</code> s.t. <code>$u=\sum_i u_i$</code>. 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. </p> <p>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 $n$.</p>