We show that the set of continuous quasinorms on a finite-dimensional linear space, after quotienting by the dilations, has a natural structure of Banach space. Our main result states that, given a finite-dimensional vector space E, the pseudometric defined in the set of continuous quasinorms \(\mathcal {Q}_0=\{\Vert \cdot \Vert :E\rightarrow \mathbb {R}\}\) as

induces, in fact, a complete norm when we take the obvious quotient \(\mathcal {Q}=\mathcal {Q}_0/\!\sim \) and define the appropriate operations on \(\mathcal {Q}\). We finish the paper with a little explanation of how this space and the Banach–Mazur compactum are related.

我们证明了在有限维线性空间上的连续拟线性集在通过膨胀商后具有Banach空间的自然结构。我们的主要结果表明，给定有限维向量空间E，在连续拟拟态\（\ mathcal {Q} _0 = \ {\ Vert \ cdot \ Vert：E \ rightarrow \ mathbb {R} \} \）为

实际上，当我们采用明显的商\（\ mathcal {Q} = \ mathcal {Q} _0 / \ !! \ sim \）并在\（\ mathcal {Q} \）上定义适当的运算时，就得出了一个完整的范数。在本文的最后，我们将对这个空间与Banach-Mazur compactum之间的关系进行一些解释。