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Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.

Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g., $$\{ f(x),e^{ix}f(x), e^{ix} f(x-1), e^{-ix}f(x-2)\}.$$ Is the volume non-zero for $0\neq f\in L^2(\mathbb R)$?

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    $\begingroup$ Of course it makes sense: the span of these vectors is a Euclidean space with inner product induced from $L^2$, thus you may consider the lengths, angles, volumes etc. The volume is non-zero if and only the elements are linearly independent. $\endgroup$
    – Fedor Petrov
    Commented Jan 15, 2019 at 20:45
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    $\begingroup$ It's a bit misleading to talk about angles in a complex inner product space. But you can still define the "volume" to be the square root of the determinant of the usual Gram matrix. $\endgroup$
    – Noam D. Elkies
    Commented Jan 15, 2019 at 20:49
  • $\begingroup$ @NoamD.Elkies: Thanks. Can we say something about the determination (zero or non zero) of Gram matrix? Any ideas or hint would be helpful to me. Thanks. $\endgroup$
    – Math Learner
    Commented Jan 15, 2019 at 21:18
  • $\begingroup$ @PiotrHajlasz: Thanks. I guess you mean determinant of Gramm matrix is zero iff vectors are LD? (If so, is there any way one can check whether the determinant is zero or non-zero)? (Checking directly vectors to be LI or LD seems to be difficult) $\endgroup$
    – Math Learner
    Commented Jan 15, 2019 at 21:44
  • $\begingroup$ The determinant of the Gramm matrix is zero iff the vectors are linearly dependent. For the proofs and properties, see notes listed in my answer, $\endgroup$
    – Piotr Hajlasz
    Commented Jan 15, 2019 at 21:54

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Given vectors $x_1,\ldots,x_n\in H$ in a Hilbert space, volume of a parallelotope can be computed as the square root of the Gramm determinant: $$ V(x_1,\ldots,x_n)=\sqrt{G(x_1,\ldots,x_n)}, $$ where $$ G(x_1,\ldots,x_n)=\det\langle x_i,x_j\rangle_{i,j=1}^n. $$ While it makes a perfect sense in the real Hilbert space, you can use it to define volume in the complex one too.

For a proof and some striking applications (in the real Hilbert space) see Sections 5.3 and 5.3 in: http://www.pitt.edu/~hajlasz/Notatki/Functional%20Analysis2.pdf.

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