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Early in the 20th century, the great mathematician David Hilbertnoticed that a number ofimportant mathematical arguments werestrcturally similar, In fact, he realized that at an appropriate level ofgenerality they could be regarded as the same. This observation, andothers like it, gave rise to a new branch of mathematics, and one of itscentral concepts was named after Hilbert, The notion of a Hilbert spacesheds light on so much of modern mathematies, from number theory toquantum mechanics, that if you do not know at least the rudiments ofHilbert space theory then you cannot claim to be a well-educatedmathematician,