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This is a tutorial introduction to the functional analysis mathematics neededin many physical problems, such as in waves in continuous media. Functionalanalysis takes us beyond finite matrices, allowing us to work with infinitesets of continuous functions. It resolves important issues, such as whether,why and how we can practically reduce such problems to finite matrixapproximations. It is, however, difficult to find a readable introduction thatis efficient and comprehensible for scientists and engineers. Here, I haveselected only the topics necessary for the most important results, but theargument is mathematically complete and self-contained. The article starts fromsets and sequences of real numbers. It then develops spaces of vectors orfunctions, introducing the concepts of norms and metrics that allow us toconsider how these can converge. Adding the inner product, it introducesHilbert spaces, and the key forms of operators that map within or between suchspaces. This leads to the concept of compact operators, which allows us toresolve many difficulties of working with infinite sets of vectors orfunctions. We then introduce Hilbert-Schmidt operators, which are compactoperators encountered extensively in physical problems, such as those involvingwaves. Finally, it introduces the eigenfunctions for major classes ofoperators, and their powerful properties, and ends with singular-valuedecomposition of operators. This article is written in a style that iscomplementary to that of standard mathematical treatments; by relegating longerproofs to a separate section, I have attempted to retain a clear narrative flowand motivation in developing the mathematical structure. Hopefully, the resultis useful to a broader readership who need to understand this mathematics,especially in physical science and engineering.

An introduction to functional analysis for science and engineering