Hilbert spaces stretch Euclid’s geometry into infinite dimensions
A Hilbert space generalizes the Euclidean plane and three-space so calculus and geometry still work in any finite or infinite dimension. David Hilbert, Erhard Schmidt, and Frigyes Riesz developed the idea in the early 1900s for PDEs, quantum mechanics, and Fourier analysis.
A Hilbert space is a vector space with an inner product, which measures lengths and angles, that is also complete: it contains enough limits for the tools of calculus to work. John von Neumann coined the name for the abstract concept behind these many uses, and the success of the approach opened a fruitful era for functional analysis. Beyond Euclidean spaces, examples include square-integrable functions, sequence spaces, Sobolev spaces of generalized functions, and Hardy spaces of holomorphic functions. They also underpin ergodic theory, the mathematics beneath thermodynamics.
The model case is ordinary three-dimensional space with the dot product, which multiplies matching coordinates and adds them. That product is symmetric, linear in its first slot, and positive definite, and any operation with those three properties counts as a real inner product. It links to geometry through the rule that x · y equals the two lengths multiplied together times the cosine of the angle between the vectors. Every finite-dimensional inner product space is automatically a Hilbert space.
Completeness in three dimensions means that a series of vectors whose lengths add up to a finite total also converges to a limit vector. Over the complex numbers, the inner product of z and w is z times the conjugate of w, and its real part is the familiar two-dimensional dot product; swapping the arguments conjugates the result, a property called Hermitian symmetry.
Exact versions of the Pythagorean theorem and the parallelogram law hold, and perpendicular projection onto a subspace drives optimization problems. Any element is pinned down by its coordinates in an orthonormal basis, much like Cartesian coordinates. When that basis is countably infinite, the space can be identified with square-summable infinite sequences, which older books simply called the Hilbert space.
Source: Hilbert space
