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In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT spaces.
A Hadamard space is defined to be a nonempty complete metric space such that, given any points x {\displaystyle x} and y , {\displaystyle y,} there exists a point m {\displaystyle m} such that for every point z , {\displaystyle z,}
The point m {\displaystyle m} is then the midpoint of x {\displaystyle x} and y : {\displaystyle y:} d = d = d / 2. {\displaystyle d=d=d/2.}
In a Hilbert space, the above inequality is equality / 2 {\displaystyle m=/2} ], and in general an Hadamard space is said to be flat if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space.