* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If the group is neither abelian nor compact, no general satisfactory theory is currently known.

If that bytecode version of the source were saved ( called packing ), it could also be executed by a much more compact version of the interpreter, called RunB ( no editor, no prettyprinter, no extraneous information included for human convenience, no debugger, ...).

If the convex hull of X is a closed set ( as happens, for instance, if X is a finite set or more generally a compact set ), then it is the intersection of all closed half-spaces containing X.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

If the lactose cannot be digested, enteric bacteria metabolize it and produce hydrogen, which, along with methane, if produced, can be detected on the patient's breath by a clinical gas chromatograph or compact solid-state detector.

If a set is compact, then it must be closed.

If S is compact but not closed, then it has an accumulation point a not in S. Consider a collection consisting of an open neighborhood N ( x ) for each x ∈ S, chosen small enough to not intersect some neighborhood V < sub > x </ sub > of a.

If a set is compact, then it is bounded.

If the spatial geometry is spherical, the topology is compact.

If the geometry of the universe is not compact, then it is infinite in extent with infinite paths of constant direction that, generally do not return and the space has no definable volume, such as the Euclidean plane.

He goes on to quote Webster further, " If the Northern States refuse, willfully and deliberately, to carry into effect that part of the Constitution which respects the restoration of fugitive slaves, and Congress provides no remedy, the South would no longer be bound to observe the compact.

Suppose V is a subset of R < sup > n </ sup > ( in the case of n = 3, V represents a volume in 3D space ) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

If we consider a general locally compact group and the connected component of the identity, we have a group extension

If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1 / 4 and 1 then M is diffeomorphic to a sphere.

If M is a non-compact complete non-negatively curved n-dimensional Riemannian manifold, then M contains a compact, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S ( S is called the soul of M .) In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to R < sup > n </ sup >.

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

If T is a compact operator, then it can be shown that any nonzero λ in the spectrum is an eigenvalue.

* If π < sub > 1 </ sub >( M ) is finite then the geometric structure on M is spherical, and M is compact.

* If π < sub > 1 </ sub >( M ) is virtually cyclic but not finite then the geometric structure on M is S < sup > 2 </ sup >× R, and M is compact.

* If π < sub > 1 </ sub >( M ) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.

The later invention of non-Euclidean geometry does not resolve this question ; for one might as well ask, " If given the axioms of Riemannian geometry, can an omnipotent being create a triangle whose angles do not add up to more than 180 degrees?

If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are ( locally ) the shortest path between points in the space.

If the torus carries the ordinary Riemannian metric from its embedding in R < sup > 3 </ sup >, then the inside has negative Gaussian curvature, the outside has positive Gaussian curvature, and the total curvature is indeed 0.

If γ: b → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L ( γ ) in analogy with the example above by

# If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT ( k ) space.

If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line ( i. e. a geodesic which minimizes distance on each interval ) then it is isometric to a direct product of the real line and a complete ( n-1 )- dimensional Riemannian manifold which has nonnegative Ricci curvature.

If q < sub > m </ sub > is positive for all non-zero X < sub > m </ sub >, then the metric is positive definite at m. If the metric is positive definite at every m ∈ M, then g is called a Riemannian metric.

* If g has signature ( n, 0 ), then g is a Riemannian metric, and M is called a Riemannian manifold.

* If G is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponential map of this Riemannian metric.

If the Ricci curvature function Ric ( ξ, ξ ) is constant on the set of unit tangent vectors ξ, the Riemannian manifold is said to have constant Ricci curvature, or to be an Einstein manifold.

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

If for each point in a connected Riemannian manifold ( of dimension three or greater ) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.

If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection.

: If ( M, g ) is a complete connected Riemannian manifold with sectional curvature K ≥ 0, then there exists a compact totally convex, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S.

In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If

If G is a Lie group and H is a Lie subgroup, then the quotient space G / H is a manifold ( subject to certain technical restrictions like H being a closed subset ) and is also a homogeneous space of G or in other words, a nonlinear realization of G. In many cases, G / H can be equipped with a Riemannian metric which is G-invariant.

If M is a Riemannian symmetric space, the identity component G of the isometry group of M is a Lie group acting transitively on M ( M is Riemannian homogeneous ).