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**Nonarchimedean functional analysis.**
*(English)*
Zbl 0998.46044

Springer Monographs in Mathematics. Berlin: Springer. 156 p. (2002).

The book is a detailed introduction to the theory of locally convex topological vector spaces over non-Archimedean fields.

The main notions and examples are given in Chapter 1. It includes various classes of spaces and the basic results like analogs of the Hahn-Banach theorem (for spaces over spherically complete fields), the Banach-Steinhaus theorem, the open mapping and closed graph theorems.

Chapter 2 is devoted to the structure of non-Archimedean Banach spaces including the problem of existence of complementary subspaces.

The duality theory is expounded in Chapter 3. In particular, the author considers admissible topologies, polarity, gives a characterization of non-Archimedean reflexive spaces, considers properties of projective and inductive limits.

The final Chapter 4 deals with topological tensor products, nuclear spaces and maps, and the Fredholm theory.

Due to the introductory character of the book, the author considers mainly spaces over spherically complete fields, the case which is more or less similar to the classical functional analysis. However this level of generality is sufficient for many applications, like the theory of non-Archimedean representation or \(p\)-adic modular forms.

The main notions and examples are given in Chapter 1. It includes various classes of spaces and the basic results like analogs of the Hahn-Banach theorem (for spaces over spherically complete fields), the Banach-Steinhaus theorem, the open mapping and closed graph theorems.

Chapter 2 is devoted to the structure of non-Archimedean Banach spaces including the problem of existence of complementary subspaces.

The duality theory is expounded in Chapter 3. In particular, the author considers admissible topologies, polarity, gives a characterization of non-Archimedean reflexive spaces, considers properties of projective and inductive limits.

The final Chapter 4 deals with topological tensor products, nuclear spaces and maps, and the Fredholm theory.

Due to the introductory character of the book, the author considers mainly spaces over spherically complete fields, the case which is more or less similar to the classical functional analysis. However this level of generality is sufficient for many applications, like the theory of non-Archimedean representation or \(p\)-adic modular forms.

Reviewer: Anatoly N.Kochubei (Kyïv)

### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |