Formally, a function belong $L^p$ if $[\int |f|^p d\mu]^{1/p} < \infty$, but it seems quite common in proofs to drop the exponent? Why?
2026-05-05 13:39:51.1777988391
Why is the exponent when working in $L^p$ spaces often dropped?
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In pure math, usually we don't care about the exact value of a non-negative real number. Rather, we care about its "categories'':
If we drop the exponent, then the math expression is simplified while these "categories'' are still preserved. On the other hand, if we care about the exact value (e.g. for proving a tight inequality), then the exponent may need to be kept, because it serves as a normalization. E.g. Consider the difference between variance and standard deviation in probability theory.