Normed versus topological groups: Dichotomy and duality

6 N. H. Bingham and A. J. OstaszewskiDefinition and Notation. For X a metric space with metric d X and π : X → X abijection the π-permutation metric is defined byWhen X is a group we will also say that d X πand for d any metric on Xd X π (x, y) := d X (π(x), π(y)).‖x‖ π := d X (π(x), π(e)),B d r (x) := {y : d(x, y) < r},suppressing the superscript for d = d X ; however, for d = d X πB π r (x) := {y : d X π (x, y) < r}.is the π-conjugate of d X . We writewe adopt the briefer notationFollowing [BePe] Auth(X) denotes the algebraic group of self-homeomorphisms (or autohomeomorphisms)of X under composition, i.e. without a topological structure. We denoteby id X the identity map id X (x) = x on X.Examples A. Let X be a group with metric d X . The following permutation metricsarise naturally in this study. (We use the notation ‖x‖ := d X (x, e X ), for an arbitrarymetric.)1. With π(x) = x −1 we refer to the π-permutation metric as the involution-conjugate, orjust the conjugate, metric and write˜d X (x, y) = d X π (x, y) = d X (x −1 , y −1 ), so that ‖x‖ π = ‖x −1 ‖.2. With π(x) = γ g (x) := gxg −1 , the inner automorphism, we have (dropping the additionalsubscript, when context permits):d X γ (x, y) = d X (gxg −1 , gyg −1 ), so that ‖x‖ γ = ‖gxg −1 ‖.3. With π(x) = λ g (x) := gx, the left-shift by g, we refer to the π-permutation metric asthe g-conjugate metric, and we writed X g (x, y) = d X (gx, gy).If d X is right-invariant, cancellation on the right givesd X (gxg −1 , gyg −1 ) = d X (gx, gy), i.e. d X γ (x, y) = d X g (x, y) and ‖x‖ g = ‖gxg −1 ‖.For d X right-invariant, π(x) = ρ g (x) := xg, the right-shift by g, gives nothing new:But, for d X left-invariant, we haved X π (x, y) = d X (xg, yg) = d X (x, y).‖x‖ π = ‖g −1 xg‖.