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In the next few lectures, we will be studying four major classes of function spaces. In decreasing order of generality, these classes are the topological vector spaces, the normed vector spaces, the Banach spaces, and the Hilbert spaces. In order to motivate the discussion of the more general classes of spaces, we will first focus on the most special class – that of (real and complex) Hilbert spaces. These spaces can be viewed as generalisations of (real and complex) Euclidean spaces such as and to infinite-dimensional settings, and indeed much of one’s Euclidean geometry intuition concerning lengths, angles, orthogonality, subspaces, etc. will transfer readily to arbitrary Hilbert spaces; in contrast, this intuition is not always accurate in the more general vector spaces mentioned above. In addition to Euclidean spaces, another fundamental example of Hilbert spaces comes from the Lebesgue spaces of a measure space . (There are of course many other Hilbert spaces of importance in complex analysis, harmonic analysis, and PDE, such as Hardy spaces , Sobolev spaces , and the space of Hilbert-Schmidt operators, but we will not discuss those spaces much in this course. Complex Hilbert spaces also play a fundamental role in the foundations of quantum mechanics, being the natural space to hold all the possible states of a quantum system (possibly after projectivising the Hilbert space), but we will not discuss this subject here.)
Hilbert spaces are the natural abstract framework in which to study two important (and closely related) concepts: orthogonality and unitarity, allowing us to generalise familiar concepts and facts from Euclidean geometry such as the Cartesian coordinate system, rotations and reflections, and the Pythagorean theorem to Hilbert spaces. (For instance, the Fourier transform is a unitary transformation and can thus be viewed as a kind of generalised rotation.) Furthermore, the Hodge duality on Euclidean spaces has a partial analogue for Hilbert spaces, namely the Riesz representation theorem for Hilbert spaces, which makes the theory of duality and adjoints for Hilbert spaces especially simple (when compared with the more subtle theory of duality for, say, Banach spaces). Much later (next quarter, in fact), we will see that this duality allows us to extend the spectral theorem for self-adjoint matrices to that of self-adjoint operators on a Hilbert space.
These notes are only the most basic introduction to the theory of Hilbert spaces. In particular, the theory of linear transformations between two Hilbert spaces, which is perhaps the most important aspect of the subject, is not covered much at all here (but I hope to discuss it further in future lectures.)