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If and U
If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.
If the United States were to punish a visiting diplomat without sufficient grounds, U. S. representatives in other countries could receive harsher treatment.
* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).
* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
Chinese immigrants were blamed for importing the opium-smoking habit to the U. S. The 1903 blue-ribbon citizens ' panel, the Committee on the Acquirement of the Drug Habit concluded, " If the Chinaman cannot get along without his dope we can get along without him.
“ The U. S. Fish and Wildlife Service-funded Ivory-billed Woodpecker searches will continue through the 2008-09 search season ,” says Laurie Fenwood, Ivory-billed Woodpecker Recovery Team Coordinator for the U. S, Fish and Wildlife Service .… If no birds are confirmed, the Cornell Lab of Ornithology will not send an organized team into the field next year.
If I manage to leave for the U. S., then I will be over there.
If a judge determines that there is sufficient evidence to believe that the defendant committed the crime, it is said that the defendant is " held to answer " or " bound over " ( in U. S. jurisdictions ).
If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.
If one considers a version of QCD with N < sub > f </ sub > flavors of massless quarks, then there is a global ( chiral ) flavor symmetry group SU < sub > L </ sub >( N < sub > f </ sub >) × SU < sub > R </ sub >( N < sub > f </ sub >) × U < sub > B </ sub >( 1 ) × U < sub > A </ sub >( 1 ).
If an electron of energy E is incident upon an energy barrier of height U ( z ), the electron wave function is a traveling wave solution,
If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level ( for metals at T = 0 kelvins ), to vacuum level.
But they soon had another US No. 2 hit with a cover of Simon & Garfunkel's " A Hazy Shade of Winter " from the soundtrack of the film Less Than Zero, and the melancholic " If She Knew What She Wants ", written by Jules Shear, reached the U. S. Top 30 and the German Top 20.
The U. S. Supreme Court explained this, in U. S. Public Workers v. Mitchell: " If granted power is found, necessarily the objection of invasion of those rights, reserved by the Ninth and Tenth Amendments, must fail.

If and .
If he had married her, he'd have been asking for trouble.
He said: `` If it's all right with you, Mr. Morgan, I'll sleep out here on the couch.
If we let them go, they won't stay away, they'll find men to ride with them and they'll be back.
He added, `` If this doesn't work out, the three of you barricade yourself in the house and talk terms with them ''.
If you've got any ideas ''.
If she did, he could stand it better in the light.
If you ever try anything without my orders I'll kill you ''.
`` If you hadn't I'd have killed you ''.
If she sensed any unusual preoccupation on the part of her mother, she did not comment upon it.
If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.
If, as she walked, her steps fumbled from time to time, she chose to ignore that omen.
If the slope grew steeper and the groves more dim, she tried not to heed.
If, when this was all over, she found the words to tell him about it, she wondered if he would ever understand.
If you take the one, you'd better take both ''.
If the bluff failed and they ran into trouble, Brannon had told the others, they would withdraw -- and he would come after his son another time.
If you don't leave this country within 3 days, your life will be taken the same as Powell's was.
If someone were to drop a match in here, this place would go up like a haystack ''.
If we was both armed, you wouldn't talk so tough ''.
`` If you spot Carmer give a yell before you move in ''.
If he wondered whether the attackers would allow him to pull away unmolested, he had his answer a moment later.
If the other pilots were worried, they did not show it.
If it were the enemy, tactically his position was correct.
If they're Japs.
If any of us miss, they can pick up the pieces.
If the turn was too tight, a barrel roll would bring them out.

If and S
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )
* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =
* If a set S is in M and S is congruent to T then T is also in M and a ( S )
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

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