If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If symptoms occur, a determination of electrolytes may be helpful in determining the underlying cause.

If this goal is met by a DBMS, then the designers and builders of the specific database can concentrate on the application's aspects, and not deal with building and maintaining the underlying DBMS.

If we compress data in a manner that assumes q ( X ) is the distribution underlying some data, when, in reality, p ( X ) is the correct distribution, the Kullback – Leibler divergence is the number of average additional bits per datum necessary for compression.

If the underlying manifold is supersymmetric, the resulting theory is a supersymmetric Yang – Mills theory.

If LEMS is caused by an underlying cancer, treatment of the malignancy usually leads to resolution of the symptoms.

If there is an underlying cause, the condition is termed " secondary intracranial hypertension ".

If this is true, then it must be the case that the wetness of water, the hardness of ice, are not essential to the underlying substance.

If care is taken not to cut too deeply, the underlying wraps will protect the bound object from being damaged by the knife.

If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism.

If short-term interest rates were expected to fall in a contango market, this would narrow the spread between a futures contract and an underlying asset in good supply.

If, on the other hand, the spread between a future traded on an underlying asset and the spot price of the underlying asset was set to widen, possibly due to a rise in short-term interest rates, then an investor would be advised to sell the spread ( i. e. a calendar spread where the trader sells the near-dated instrument and simultaneously buys the future on the underlying ).

If the computational Diffie – Hellman assumption holds in the underlying cyclic group, then the encryption function is one-way.

If no underlying cause can be identified, the condition is considered idiopathic.

If the concept of belief is incoherent, then any attempt to find the underlying neural processes that support it will fail.

If dwarfism is caused by a medical disorder, the person is referred to by the underlying diagnosed disorder.

If the underlying problem is well-posed, there should be a stable algorithm for solving it.

If properly treated, typical cases of surgically correctable peritonitis ( e. g., perforated peptic ulcer, appendicitis, and diverticulitis ) have a mortality rate of about < 10 % in otherwise healthy patients, which rises to about 40 % in the elderly, and / or in those with significant underlying illness, as well as in cases that present late ( after 48 hours ).

If natural selection is the primary error correction mechanism that prevents random mutations on the Y chromosome, then fertility treatments for men with abnormal sperm ( in particular ICSI ) only defer the underlying problem to the next male generation.

If the strike is K and maturity time is T, and the buyer exercises the put at a time t, the buyer can expect to receive a payout of K-S ( t ), if the price of the underlying S ( t ) at that time is less than K. The exercise t must occur by time T ; precisely what exact times are allowed is specified by the type of put option.

If the underlying stock's market price is below the option's strike price when expiration arrives, the option owner ( buyer ) can exercise the put option, forcing the writer to buy the underlying stock at the strike price.

If g is the metric on this manifold, one defines the action S ( g ) as

If the base manifold is four-dimensional, the Kaluza – Klein manifold P is five-dimensional.

# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.

If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are ( more or less ) Kac – Moody algebras.

* If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f ( x )

If M is an open subset of R < sup > n </ sup >, then M is a C < sup >∞</ sup > manifold in a natural manner ( take the charts to be the identity maps ), and the tangent spaces are all naturally identified with R < sup > n </ sup >.

If the manifold is the surface of the Earth,

If the range of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.

If the manifold M is smooth ( respectively analytic )--- that is, the change of coordinates are smooth ( respectively analytic )--- then one can make sense of the notion of smooth ( respectively analytic ) vector fields.

If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold.

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 the field is a real vector field, then the target manifold is isomorphic to R < sup > 3 </ sup >.

If is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not null.

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 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 compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.

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 the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.