Inner product space

An inner product space is a real or complex vector space equipped with a positive-definite inner product. The inner product defines lengths and angles and induces a norm.

Vector space: A set with addition and scalar multiplication satisfying the vector space axioms. Inner product spaces add a geometric pairing to this underlying algebraic structure.
Pythagorean theorem: A relation stating that the squared hypotenuse length of a right triangle equals the sum of the squared leg lengths. Orthogonal vectors satisfy this identity through the inner-product norm.
Hilbert space: An inner product space that is complete under the norm induced by its inner product. Completeness makes infinite-dimensional inner-product geometry suitable for analysis.
Banach space: A normed vector space that is complete under the metric induced by its norm. A Banach space need not have a norm arising from any inner product.
Inner product: A positive-definite sesquilinear form, or bilinear form over the reals, on a vector space. This pairing supplies the lengths and angles associated with the space.
Orthogonal projection: A linear map sending vectors to their closest points in a specified subspace under an inner-product norm. It uses inner products to split vectors into parallel and perpendicular components.
Fourier series: A representation of a periodic function as a sum of sinusoidal components. Orthogonality of sine and cosine functions determines Fourier coefficients.
Minkowski space: A real vector space with a nondegenerate indefinite bilinear form used in special relativity. Its form permits negative squared lengths, unlike a positive-definite inner product.
Normed vector space: A vector space equipped with a function assigning each vector a nonnegative length that satisfies the norm axioms. Every inner product induces a norm, placing these spaces within normed-space theory.
Gram–Schmidt process: An algorithm that converts a linearly independent list into an orthogonal or orthonormal list spanning the same subspace. It constructs orthonormal coordinates from independent vectors.

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