Timeline for Cone in a metric space
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 19, 2022 at 4:36 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the dead link
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| May 16, 2011 at 14:39 | comment | added | Todd Trimble | The construction given by KConrad can be considered as the enriched Yoneda embedding of a metric space, when seen as a category enriched in the symmetric monoidal closed structure on the preorder of real numbers (with order opposite to the standard order) equipped with the symmetric monoidal product given by addition. This is the beginning observation of Lawvere's famous paper on metric spaces and enriched category theory. | |
| May 16, 2011 at 11:42 | comment | added | KConrad | and ||g_x - g_y|| = d(x,y). Thus (X,d) embeds isometrically into B(X) by sending each x in X to the function g_x. This is not "canonical" since it depends on a choice of a. If the metric d is bounded on X then f_x is already in B(X) so we can drop the whole business with a: sending x to f_x is an isom. embedding of X into B(X). | |
| May 16, 2011 at 11:42 | comment | added | KConrad | Embedding a metric space (X,d) isometrically into a Banach space can be done in a simpler way, as in Lang's Undergraduate Analysis, exercise 3 in Chapter VI, Section 2. For each x in X, let f_x : X --> R by f_x(y) = d(x,y). This is a continuous function of y (Lang's exercise doesn't mention the continuity), although f_x need not be bounded. Fix a in X and define g_x = f_x - f_a (i.e., g_x(y) = d(x,y) - d(a,y)). Each g_x : X --> R is continuous and is bounded by the triangle inequality. The space B(X) of continuous bounded functions X --> R is a Banach space using the sup norm [continued] | |
| Jan 24, 2010 at 21:59 | comment | added | Yemon Choi | @Pete: I think it might be a left adjoint, IIRC; I remember seeing this mentioned in a talk of Godefroy some years ago. That is, every nonexpansive map from a (pointed?) metric space X into a Banach space E, extends canonically to a contractive linear map from F(X) into E, where F(X) is this Arens-Eells widget | |
| Jan 24, 2010 at 12:45 | comment | added | Pete L. Clark | In what sense is this embedding canonical? | |
| Jan 24, 2010 at 10:51 | history | answered | Aryeh Kontorovich | CC BY-SA 2.5 |