* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

Conversely, a complete subset of a metric space is closed.

Conversely, every Polish space is homeomorphic to a G < sub > δ </ sub >- subset of the Hilbert cube.

Conversely, words such as banjee, while well-established in a subset of gay society, have never made the transition to popular use.

Suppose S is an immersed submanifold of M. If the inclusion map i: S → M is closed then S is actually an embedded submanifold of M. Conversely, if S is an embedded submanifold which is also a closed subset then the inclusion map is closed.

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

Conversely, given central idempotents a < sub > 1 </ sub >,..., a < sub > n </ sub > in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra < sub > 1 </ sub >,…, Ra < sub > n </ sub >.

Conversely in rings with unity, any simple left R module arises this way.

** Conversely, if ƒ: I → R is absolutely continuous and thus differentiable almost everywhere, and satisfies | ƒ ′( x )| ≤ K for almost all x in I, then ƒ is Lipschitz continuous with Lipschitz constant at most K.

Conversely, it is possible to 2-colour a K < sub > 5 </ sub > without creating any monochromatic K < sub > 3 </ sub >, showing that R ( 3, 3 ) > 5.

Conversely, if R is a symmetric relation over a set X, one can interpret it as describing an undirected graph with the elements of X as the vertices and the pairs in R as the edges.

Conversely, an exhaustive firing test of the M1918 Chauchat in. 30-06 was also carried out in 1994 near Chambersburg, Pennsylvania, by R. Keller and W. Garofalo.

Conversely, the following characterization of derived functors holds: given a family of functors R < sup > i </ sup >: A → B, satisfying the above, i. e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, R < sup > i </ sup >( I )= 0 for every positive i, then these functors are the right derived functors of R < sup > 0 </ sup >.

Conversely, given a Boolean ring R, we define the join operation by a ' b = a + b + ab, and the meet operation as multiplication.

Conversely, if this identity holds in a ring R for all pairs of elements a and b of the ring, then R is commutative.

Conversely, Psychology defines bottom-up processing as an approach wherein there is a progression from the individual elements to the whole.

Conversely, given a commutative diagram, it defines a poset category:

Conversely, a system of n quantities v < sup > i </ sup > that transform like the coordinates x < sup > i </ sup > on V defines a contravariant vector.

Thus, b < sub > q </ sub > is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

Conversely the building can be recovered from the BN pair, so that every BN pair canonically defines a building.

where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by

Conversely, any operator satisfying the above properties defines a connection on E and a connection in this sense is also known as a covariant derivative on E.

Conversely, an Ehresmann connection H ⊂ TP ( or v: TP → V ) on P defines a principal G-connection ω if and only if it is G-equivariant in the sense that.

Conversely, any such one form defines ( via pullback ) a G-equivariant horizontal 1-form on P, and the space of principal G-connections is an affine space for this space of 1-forms.

Conversely, a crossed module with surjective boundary defines a central extension.

Conversely, singly even dimensional manifolds have a skew-symmetric nondegenerate bilinear form on their middle dimension ; if one defines a quadratic refinement of this to a quadratic form ( as on a framed manifold ), one obtains the Arf invariant as a mod 2 invariant.

Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel.

Conversely, function application is considered to be left-associative, so that is equivalent to.

Conversely, sociology of law may help comparative law answer questions, such as: How do regulations in different legal systems really function in the respective societies?

Conversely, in functional code, the output value of a function depends only on the arguments that are input to the function, so calling a function f twice with the same value for an argument x will produce the same result f ( x ) both times.

Conversely, given a harmonic function, it is the real part of an analytic function, ( at least locally ).

Conversely most ( but not all ) of the ASCII " control characters " have some control function in isolation, therefore are not escape characters.

Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally.

Conversely, those who wish to function primarily as clinical ED physicians choose the CCFP route of certification.

Conversely, warming of an affected extremity may indicate a disruption of the body's normal thermoregulatory vasoconstrictor function, which may sometimes indicate underlying CRPS.

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is ( locally ) a space form.

Conversely, any equation can take the canonical form f ( x ) = 0, so equation solving is the same thing as computing ( or finding ) a root of a function.

Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d ( p, q ) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function ( taken in a suitable manner ).

Conversely, by the inverse function theorem, there can be at most one value for σ that, when applied as an input to, will result in a particular value for C.

Conversely, culture decline is a function of the feckless mixture of Aryan, with lesser ' animalistic ,' blood ...

Conversely, if it is too wet, mud cakes in the machinery and moisture weighs down the material, making it impossible for the equipment to function without picking up large quantities of mud.

Conversely, a single email address may be an alias and have a distribution function to many mailboxes.

Conversely, there may be levels of production where marginal cost is higher than average cost, and average cost is an increasing function of output.

Conversely, a function u: E < sub > 1 </ sub > → E < sub > 2 </ sub > defined on a basis of E < sub > 1 </ sub > can be uniquely extended to a linear function.