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The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces (p ≥ 1), and inner product spaces.
Cosine similarity. In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not ...
The triangle inequality and the fact that (,) = for all points x derive from the properties of composition and identity in an enriched category. Since R ∗ {\displaystyle R^{*}} is a poset, all diagrams that are required for an enriched category commute automatically.
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space[1][2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive ...
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...
Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that +, with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths :
ℓ ∞ , {\displaystyle \ell ^ {\infty },} the space of bounded sequences. The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: Define the -norm:
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...