1 The \(l_p\) Spaces

Functional analysis studies spaces whose points are themselves functions or sequences, and this first section builds the examples the rest of the course argues about.

For a fixed \(p\geq 1\), the space \(l_p\) consists of the infinite sequences whose \(p\)-th powers have a convergent sum. Whether that is a sensible object at all turns on one question: is the sum of two such sequences again one? The answer is Minkowski’s inequality, and it is not obvious — it is proved here from Hölder’s inequality, which is proved in turn from Young’s.

That chain is the whole content of the section, and it is worth keeping in view: each inequality exists to make the next one possible, and the last of them is precisely the triangle inequality for the \(l_p\) norm.

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