Sentence examples for be an arbitrarily given from inspiring English sources

Let (kin P) be an arbitrarily given vector, then ((X,F)) is a Hausdorff space.

Let V be a real vector space, let Y be a real normed space, and let (f : V to Y) be an arbitrarily given mapping.

end{aligned} Let (U=bigcap_{i=1}^{n}r_{i}V(rho_{i})) be an arbitrarily given neighborhood of the origin in X.

Let (p notin{ 1, 2, 3 }) and (theta> 0) be real constants, let X, Y be real normed spaces, and let (f : X to Y) be an arbitrarily given mapping.

Let (a neq1) and (b = 1) be positive rational numbers, let (phi: X backslash{ 0 } to(Z,N_{Z})) be a mapping satisfying either (5) for all (x in X backslash{ 0 }) or (6) for all (x in X backslash{ 0 }), and let (f : X to(Y,N_{Y})) be an arbitrarily given mapping.

Let (a neq1) and (b = 1) be positive rational numbers, let Y and Z be real normed spaces, let (phi: X backslash{ 0 } to Z) be a mapping satisfying either (11) for all (x in X backslash{ 0 }), or (12) for all (x in X backslash{ 0 }), and let (f : X to Y) be an arbitrarily given mapping.

Assume that the real-valued constants α, β are given with ((alpha- 1)^{2} - 4 beta> 0) and ε is an arbitrarily given positive constant.

Assume that the real-valued constants α and β are given with (alphaneq1), (beta= frac{ alpha- 1)^{2}{{4}) and ε is an arbitrarily given positive constant.

Assume that the real-valued constants α and β are given with ((alpha- 1)^{2} - 4 beta< 0) and ε is an arbitrarily given positive constant.

Let ({u_{n}}) be a c-sequence in P. If (beta in {P}) is an arbitrarily given vector, then ({beta u _{n}}) is a c-sequence.

Suppose that (kin P) is an arbitrarily given vector and (xll c) for any (thetall c), then we have (kxll c) for any (thetall c).