Hilbert space
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A Hilbert space is a mathematical space where you can measure distances and angles between points, and it's complete in the sense that every sequence of points that gets progressively closer together actually converges to a point within that space. This structure is fundamental to quantum mechanics and many areas of physics and mathematics because it provides the right framework for describing how systems evolve and how quantities can be measured.
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The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space.
The mathematical concept of a Hilbert space generalizes the notion of Euclidean space. It extends the methods of Euclidean geometry and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces of any finite or infinite dimension. A Hilbert space is an abstract vector space, and it has the additional structure of an inner product that allows length and angle to be measured. Finally, Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used.