mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be...

The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space...

areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated...

admits no finite subcover. It is complete but not totally bounded. For an ordered space (X, <) (i.e. a totally ordered set equipped with the order topology)...

metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For...

of the topology used, in a particular theory. Compactly embedded Totally bounded space page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators...

definition of boundedness can be weakened a bit; E {\displaystyle E} is bounded if and only if every countable subset of it is bounded. A set is bounded if and...

resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated...

the measure space ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} is called a finite measure space or a totally finite measure space. For any measurable...

n-space, every ball is bounded by a hypersphere. The ball is a bounded interval when n = 1, is a disk bounded by a circle when n = 2, and is bounded by...

space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded....

discrete uniform or metric space is complete. Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is...

Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space. Hausdorff distance Kuratowski convergence Vietoris topology Hemicontinuity...

converges (i.e. has a limit point in X). Totally bounded (or Precompact). A uniform space X is totally bounded if for each entourage E ⊂ X × X there is...

subset of a totally ordered set X is totally ordered for the restriction of the order on X. The unique order on the empty set, ∅, is a total order. Any...

vector space X {\displaystyle X} is said to have the Heine–Borel property (R.E. Edwards uses the term boundedly compact space) if each closed bounded set...

convergent sequence {xn} is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly convergent...

complete and totally bounded. A Fréchet–Montel space is a Fréchet space that is also a Montel space. A separable Fréchet space is a Montel space if and only...

topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space X ′ . {\displaystyle X^{\prime }...

applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used...

functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone...

lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below)...

Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets...

into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which...

space. In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded. In a complete locally convex space...

as path isometry. Asymptotic cone Autoparallel the same as totally geodesic. Banach space Barycenter, see center of mass. Bi-Lipschitz map. A map f :...

functionals on a preordered vector space X {\displaystyle X} that map every order interval into a bounded set is called the order bound dual of X {\displaystyle...

into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. every sequence in such a bounded set has a subsequence that is Cauchy...

something totally other than ourselves—"something numinous"—and the conflicting desire for a beauty that makes us feel no longer "lost in space", but at...

be of bounded variation precisely if its total variation is finite. The total variation is a norm defined on the space of measures of bounded variation...