We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space \(\ell _2\) and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

中文翻译：

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什么时候映射保持半内积线性？

我们观察到，尊重固定半内积的相同维度的有限维赋范空间之间的每个映射必须自动是线性等距。此外，我们构建了希尔伯特空间\(\ell _2\)的均匀平滑重归一化和作用在其上的连续注入，该注入尊重半内积，但它是非线性的。这表明，即使在均匀平滑的额外假设下，前一个结果也不会立即扩展到无限维。