A two variable real function $F(y,x)$, defined on $[c,d] \times [a,b]$, can be thought of as a linear map between functions of different domains by: $$ g(y) = \int^a_bF(y,x)f(x)dx $$ This has very obvious analogies with matrix transformations on discrete vector spaces $$ g_i = \sum_j^n F_{ij}f_j $$ Is there an analogous 'determinant' for $F(x,y)$ like there is with $\det (F_{ij})$?
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$\begingroup$ I don't think we quite have a determinant, however in some cases, you can view this as a case of Hilbert-Schmidt integral operator, which is a compact operator, and such operators have quite reasonable "linear algebra" theory. All this lives in the realm of functional analysis. $\endgroup$Wojowu– Wojowu2025-10-22 13:21:31 +00:00Commented Oct 22 at 13:21
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$\begingroup$ I don't know the answer, but this reminds me of QFT and the similar analogies physicists make with linear algebra. $\endgroup$Malkoun– Malkoun2025-10-22 13:30:17 +00:00Commented Oct 22 at 13:30
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5$\begingroup$ There is a generalization of determinant; is is called "Fredholm determinant" (see Wikipedia). $\endgroup$Alexandre Eremenko– Alexandre Eremenko2025-10-22 15:10:32 +00:00Commented Oct 22 at 15:10
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$\begingroup$ You can define singular values of compact operators; those could replace the determinant for some of its uses. $\endgroup$Federico Poloni– Federico Poloni2025-10-23 12:44:00 +00:00Commented Oct 23 at 12:44
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