*(English)*Zbl 0804.46030

[For the entire collection see Zbl 0746.00071.]

This paper gives a survey of some results (due to, e.g., the authors, Behrends, Cambern, and many others) connected with the Banach-Stone theorem and with corresponding characterizations of isometries and small bound isomorphisms on various function spaces and function algebras. Here are the contents.

Section 3: Banach-Stone theorem, Section 4: (injective, surjective) isometries between subspaces of spaces of continuous functions, Section 5: isometries between Banach function spaces and algebras (5.1 differentiable functions, 5.2 absolutely continuous functions, 5.3 Lipschitz functions, 5.4 general Banach function spaces, 5.5 semisimple commutative Banach algebras), Section 6: isometries of spaces of vector valued functions (6.1 $E$-valued continuous functions, 6.2 injective tensor products of Banach spaces, 6.3 $C\left(K\right)$-modules, 6.4 vector valued ${\text{weak}}^{*}$-continuous functions, 6.5 vector valued analytic functions $[{H}^{\infty},{H}^{1}]$), Section 7: isomorphisms with small bounds (7.1 spaces of continuous functions, 7.2 injective isomorphisms of spaces of continuous functions, 7.3 spaces of differentiable functions, 7.4 spaces of continuous vector valued functions, 7.5 nonlinear Banach- Stone theorem).

##### MSC:

46E15 | Banach spaces of continuous, differentiable or analytic functions |

46J10 | Banach algebras of continuous functions, function algebras |

46J15 | Banach algebras of differentiable or analytic functions, ${H}^{p}$-spaces |

46M05 | Tensor products of topological linear spaces |

46B04 | Isometric theory of Banach spaces |

46B25 | Classical Banach spaces in the general theory of normed spaces |

46E40 | Spaces of vector- and operator-valued functions |

46H25 | Normed modules and Banach modules, topological modules |