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 the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.

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 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 there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If action is indicated, what kind of action is relevant to the problem??

If, as I suspect, the problem is largely of the second sort, then development of a theory better able to handle tone will result automatically in better theory for all phonologic subsystems.

If the problem is enlarged to require a complete coverage of feed states, Af operations are needed by the dynamic program and Af by the direct search.

If high accuracy is required in preflight leveling, it is usually necessary to integrate or doubly integrate the accelerometer outputs ( this also minimizes the noise problem ).

If he foresaw any problem because of the quality of the hymen, it was recommended that simple procedures be undertaken at once to incise the hymen or, preferably, to dilate it.

`` If you substitute ' atom ' for ' angel ', the problem is not insoluble, given the metallic content of the pin in question ''.

If we allow only quantifiers, it becomes the Co-NP-complete tautology problem.

If we allow both, the problem is called the quantified Boolean formula problem ( QBF ), which can be shown to be PSPACE-complete.

If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP.

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete ( since otherwise NP = co-NP ).

If a problem is suspected then a more comprehensive test using a leak-down tester can locate the leak.

If I told you my son's age, then there would no longer be two unknowns ( variables ), and the problem becomes a linear equation with just one variable, that can be solved as described above.

If at least 50 cars had been built, sportscars like the GT40 and the Lola T70 were allowed, with a maximum of 5. 0 L. John Wyer's revised 4. 7 litre ( Bored to 4. 9 litres, and o-rings cut and installed between the deck and head to prevent head gasket failure, a common problem found with the 4. 7 engine.

If the open string is in tune, but sharp or flat when frets are pressed, the bridge saddle position can be adjusted with a screwdriver or hex key to remedy the problem.

If the status code indicated a problem, the user agent might display the reason phrase to the user to provide further information about the nature of the problem.

If never, the problem is likely to be physiological ; if sometimes ( however rarely ), it could be physiological or psychological.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

If the example with the colored bricks above is viewed as an unbounded knapsack problem, then the solution is to take three yellow boxes and three grey boxes.

(...) If someone asks me how I can sleep at night knowing that my arms have killed millions of people, I respond that I have no problem sleeping, my conscience is clean.