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I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant.

Let $X$ be a Banach space with norm $|\cdot|$. When $x \in X$ and $W \subset X$ is a closed vector space not containing $x$, write $\text{dist}(x, W) = \inf\{|x - w| : w \in W \}$. Write $\langle x_1, x_2, \cdots, x_k \rangle$ for the finite dimensional subspace of $X$ spanned by $x_1, \cdots x_k \in X$.

For $n \geq 1$ and any $n$-tuple of vectors $x_1, \cdots, x_n \in X$, define $$ V_n(x_1, \cdots, x_n) = |x_n| \cdot \text{dist}(x_{n-1}, \langle x_n \rangle) \cdot \text{dist}(x_{n-2}, \langle x_{n-1}, x_n \rangle ) \cdots \text{dist}(x_1, \langle x_2, \cdots, x_n \rangle ) $$

Notice that when $X$ is a Hilbert space, $V_n(x_1, \cdots, x_n) = |x_1 \wedge \cdots \wedge x_n|$ is precisely the (unsigned) volume of the parallelopiped spanned by $x_1, \cdots, x_n$.

I found this definition in the monograph of Z. Lian and K. Lu, where they use it to prove their version of the (Oseledets) Multiplicative Ergodic Theorem for asymptotically compact cocycles on Banach spaces. There they prove some properties for $V_n$, but they don't give any independent references. I think there ought to be, as this seems to me a natural way to define volume on normed vector spaces in general.

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Grothendieck developed a Fredholm determinant theory for Hilbert spaces that Pisier generalized to Banach spaces and used for his study of weak Hilbert spaces. You can read about that in Pisier's book MR1036275 Reviewed Pisier, Gilles The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. xvi+250 pp. ISBN: 0-521-36465-5; 0-521-66635-X – Bill Johnson Sep 20 '13 at 16:55

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