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
\begin{equation} c\_{(F,|\cdot|)}|x|\leq\|x\|\leq C\_{(F,|\cdot|)}|x| \end{equation}
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 $X$?



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


