# Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a norm for any $n = 2, 3, 4, \dots$ ? It should be noted that there are several classes of proper E. A) For $\mathbb K = \mathbb R, \mathbb C$ any $E$ is proper. (This is proved in textbooks with the use of the Hahn-Banach theorem.) B) For any set $X$ and any complete normed field $\mathbb K$ and $q \in [1, + \infty]$ the Banach space $l_q(X,\mathbb K)$ is proper. C) Any non-Archimedean Banach space over a spherically complete, locally compact non-Archimedean normed field $\mathbb K$ (e.g. over $\mathbb Q_p$ - field of $p$-adic numbers) is proper. The proper Banach spaces $E$ are used for a construction of tensor Banach algebras (and a proper tensor-Banach functor).

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