# Name for certain property of equivalent norms on finite-dimensional subspaces of a Banach space

Let $$X=(X,\|\cdot\|)$$ be a Banach space and suppose that $$F\subset X$$ is a finite-dimensional subspace. There is then an equivalent norm $$|\cdot|$$ on $$F$$ such that $$|\cdot|$$ is induced by an inner product on $$F$$ (i.e. $$|\cdot|$$ will satisfy the parallelogram law) and it follows that $$$$c_{(F,|\cdot|)}|x|\leq\|x\|\leq C_{(F,|\cdot|)}|x|$$$$ for some constants $$c,C>0$$ and for all $$x\in F$$. Here is my main question: is there a name for the following property?

There exists $$M\geq 1$$ such that for every finite-dimensional subspace $$F\subset X$$, there is an equivalent norm $$|\cdot|$$ on $$F$$ that is induced by an inner product on $$F$$ and is such that $$1\leq \frac{C}{c}\leq M$$.

Clearly, any Hilbert space has this property by taking $$M=1$$ and $$|\cdot|=\|\cdot\|$$. Are there examples of non-Hilbert spaces that have this property? Is this property related somehow to the type/cotype of $$X$$?

• The property has no quantifier on $c,C$. Probably you define $c,C$ as functions of $(F,|\cdot|)$ in the first paragraph but you could be more explicit.
– YCor
Dec 4, 2021 at 17:08
• Yes, this is exactly what I meant. Dec 4, 2021 at 17:12
• This is finite representability. Please see the definition in this question: mathoverflow.net/questions/151758/… Dec 4, 2021 at 17:48
• See also books.google.ca/… Dec 4, 2021 at 17:55

## 1 Answer

Your condition implies that $$X$$ is isomorphic to a Hilbert space with isomorphism constant at most $$M^2$$. The distance condition implies that both type 2 constant and cotype 2 constant of $$X$$ is bounded by $$M$$. By Kwapien theorem the Banach-Mazur distance of $$X$$ to a Hilbert space is bounded by type 2 constant times cotype 2 constant.

As a reference see e.g., Albiac-Kalton.

• Got it. I had a feeling something like this was the case. Thanks, Bunyamin Dec 4, 2021 at 18:05
• That is a perfectly fine answer, but using Kwapien's Theorem is a bit of an overkill. It is an elementary compactness argument that the condition implies that X is isomorphic to a Hilbert space with isomorphism constant at most M. Dec 4, 2021 at 22:19