Suppose that K is a field ( for example, the real numbers ) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars.

Suppose that W is a subset of V.

Suppose V and W are vector spaces over the field K. The cartesian product V × W can be given the structure of a vector space over K by defining the operations componentwise:

Suppose that exactly two of U, V, W are 0.

Suppose we have a connected graph G = ( V, E ), The following statements are equivalent:

Suppose now that an attenuating feedback loop applies a fraction β. V < sub > out </ sub > of the output to one of the subtractor inputs so that it subtracts from the circuit input voltage V < sub > in </ sub > applied to the other subtractor input.

where g ( w ) = f ( w ) v. Suppose that V is finite dimensional.

Suppose V is a model of ZFC.

Suppose that ( V, ω ) and ( W, ρ ) are symplectic vector spaces.

First proof: Suppose forms a basis of ker T. We can extend this to form a basis of V:.

Suppose M is a compact smooth manifold, and a V is a smooth vector bundle over M. The space of smooth sections of V is then a module over C < sup >∞</ sup >( M ) ( the commutative algebra of smooth real-valued functions on M ).

The covariant derivative can also be constructed from the Cartan connection η on P. In fact, constructing it in this way is slightly more general in that V need not be a fully fledged representation of G. Suppose instead that that V is a (, H )- module: a representation of the group H with a compatible representation of the Lie algebra.

Suppose that V is only a representation of the subgroup H and not necessarily the larger group G. Let be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

Suppose we define V to be the set of variables on which the functions f and g operate.

Suppose V is a vector space over K, a subfield of the complex numbers ( normally C itself or R ).

Suppose the subspaces U and V are the range and null space of P respectively.

Suppose that V is a vector space with a nondegenerate bilinear form (·,·).

Suppose Af is defined in the sub-interval Af.

Suppose they both had ventured into realms which their colleagues thought infidel: is this the way gentlemen settle frank differences of opinion??

Suppose, says Dr. Lyttleton, the proton has a slightly greater charge than the electron ( so slight it is presently immeasurable ).

Suppose it is something right on the planet, native to it.

Suppose there is a program

Suppose there is a chain at 1A, 2A, 3A, and 4A, along with another chain at 6A and 7A.

If two players tie for minority, they will share the minority shareholder bonus. Suppose Festival is the chain being acquired.

Alex is the majority shareholder, and Betty is the minority shareholder. Suppose now that Worldwide is the chain being acquired.

Suppose that R ( x, y ) is a relation in the xy plane.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Suppose that a car is driving up a tall mountain.

Suppose that the car is ascending at 2. 5 km / h.

Suppose the vector field describes the velocity field of a fluid flow ( such as a large tank of liquid or gas ) and a small ball is located within the fluid or gas ( the centre of the ball being fixed at a certain point ).

Suppose that F is a partial function that takes one argument, a finite binary string, and possibly returns a single binary string as output.

Suppose, says Searle, that this computer performs its task so convincingly that it comfortably passes the Turing test: it convinces a human Chinese speaker that the program is itself a live Chinese speaker.

; Dennett's reply from natural selection: Suppose that, by some mutation, a human being is born that does not have Searle's " causal properties " but nevertheless acts exactly like a human being.

Suppose that is a complex-valued function which is differentiable as a function.

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension / compression of the spring.

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Suppose that u and v are real-differentiable at a point in an open subset of, which can be considered as functions from to.

Suppose that u and v satisfy the Cauchy – Riemann equations in an open subset of R < sup > 2 </ sup >, and consider the vector field

Suppose M is a C < sup > k </ sup > manifold ( k ≥ 1 ) and x is a point in M. Pick a chart φ: U → R < sup > n </ sup > where U is an open subset of M containing x.

An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: Suppose X is a subset of R well-ordered by ≤.

Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element.

* 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.

Suppose that L is a lattice of determinant d ( L ) in the n-dimensional real vector space R < sup > n </ sup > and S is a convex subset of R < sup > n </ sup > that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S.

Suppose G is an ordered abelian group, meaning an abelian group with a total ordering "<" respecting the group's addition, so that a < b if and only if a + c < b + c for all c. Let I be a well-ordered subset of G, meaning I contains no infinite descending chain.

Suppose U is an open subset of the complex plane C, f: U → C is a holomorphic function and the closed disk

Suppose U is a simply connected open subset of the complex plane, and a < sub > 1 </ sub >,..., a < sub > n </ sub > are finitely many points of U and f is a function which is defined and holomorphic on U

Suppose f is an analytic function defined on an open subset U of the complex plane.

Suppose that the alternatives can be partitioned into subsets, where each subset has its own cost function, and each alternative belongs to only one subset.

Suppose that is an open subset of the complex plane, that is a holomorphic function on, and that is a closed curve in.

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.

Suppose Ω is an open, connected and convex subset of the Euclidean space R < sup > 2 </ sup > with smooth boundary ∂ Ω and suppose that D is the unit disk.