Does anyone know if there is an English translation of Hilbert's: "Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig, 1912". ??
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this 2014 overview lists the known translations in English of Hilbert's books, the 1912 book in the OP is not among them:
Yes, there is a translation, but not an official one and only of the "Erste Mitteilung", not the whole book: The pdf
contains "Foundations of a General Theory of Linear Integral Equations" (starting on p.55 (59 in the pdf)). This is a translation of the paper
Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Erste Mitteilung Nachrichten der Wissenschaftlichen Gesellschaft zu Göttingen, Math.-phys. Kl. (1904),49-91".
I haven't seen translation of the five other "Mitteilung" under the same title (but I haven't seen version of the second, third and sixth Mitteilung even in German…).
The other two papers in the pdf are Fredholm's "On a Class of Functional Equations" and Schmidt's "On the Theory of Linear and Nonlinear Integral Equations. Part I: The Expansion of Arbitrary Functions by Prescribed Systems."