Vector algebra and calculus are methods normally used in the two-dimensional Euclidean plane and three-dimensional space. In Hilbert spaces, these methods can be used with any finite or infinite number of dimensions. A Hilbert space is a vector space that has the structure of an inner product that allows length and angle to be measured. Hilbert spaces also have to be complete, which means that enough limits have to exist for calculus to work.
The earliest Hilbert spaces were studied in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. John von Neumann first came up with the name "Hilbert Space". Hilbert space methods made a big difference to functional analysis.
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