Usually, when studying the applications and results of a theory, it becomes clear why it was interesting to define it in a certain way. However, I'm currently beginning my studies in functional analysis. I see $l^p$ spaces as a generalization of Euclidean space, for vectors in an infinitely-dimensional space. But I would like to understand the construction of this space in a logical manner, comprehending the chronology of definitions, where it originated, who defined it, and everything else. Does anyone have a good recommendation for a history on this?
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3$\begingroup$ History is messy and chronology is rarely logical or explanatory. Hilbert defined infinite dot product in connection with integral equations, but did not consider spaces with it. Schmidt defined what we now call $l^2$ in 1907 to study infinite systems of linear equations. Then Riesz went from $2$ to $p$ around 1910 simply because Schmidt's theory could be generalized (he considered even more general $L^p$). You can read details in Dieudonne's History of Functional Analysis, VI.1. $\endgroup$– ConifoldOct 22 at 8:55
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As I recall, there are extensive historical notes in both:
Dunford & Schwartz, Linear Operators (Interscience, 1958)
M. M. Day, Normed Linear Spaces (Springer, 1958)
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2$\begingroup$ See also Origin and first uses of $\ell_p$ norms? (mathoverflow). Also likely useful are The development of function spaces with particular reference to their origins in integral equation theory by Michael Bernkopf (1966) AND A study of Maurice Fréchet: III Fréchet as analyst, 1909-1930 by Angus E. Taylor (1987). $\endgroup$ Oct 23 at 15:35