Sentence examples for there exists a net from inspiring English sources

If, then there exists a net in such that.

Indeed, if, then there exists a net in such that.

So there exists a net ( x i ) ∈ X ∗ such that D ( a ) = lim i δ x i σ ( a ), ( a ∈ A ).

Then is l.s.c. at if and only if for any and any net in converges to, there exists a net such that for all and.

Then, is l.s.c. at if and only if for any and for any net in converging to, there exists a net in such that for each and converges to.

By Theorem 2.2, there exists a net ((a_i,b_i)_i) in (Aoplus _pB), such that (varphi circ (phi,0)(a_i,b_i longrightarrow 1) and begin{aligned} Vert (a,b).(a_i,b_i -(a_i,b_i). a,b Vert longrightarrow 0, Vert alpha.(a_i,b_i -(a_i,b_i).alpha Vert longrightarrow 0, end{aligned}for all ((a,b in Aoplus _pB) and (alpha in mathfrak {A}).

If, then there exist a net such that and a net in such that for all.

Since B is u.s.c. with compact values, by Lemma 2.2, there exists a subset net of ({z_{alpha}}), denoted again by ({z_{alpha}}), such that (z_{alpha}rightarrow z_{0}in B(x_{0})).

Since (F cdot,y,cdot)) is u.s.c. with compact valued, by Lemma 2.2, there exists a subset net of ({v_{alpha }}), denoted again by ({v_{alpha}}), such that (v_{alpha }rightarrow v_{0}in F(x_{0},y,z)).

Then, there exists a bounded net ((a_i)_i) in A, such that (varphi circ phi (a_i longrightarrow 1), (Vert aa_i-a_iaVert longrightarrow 0) and (Vert alpha.a_i-a_i.alpha Vert longrightarrow 0) for all (ain A, alpha in mathfrak {A}).

By Mazur's theorem, (0in overline{T(B)}^w=overline{T(B)}^{Vert.Vert }.) Therefore, there exists a bounded net ((a_i)_i) in A, such that (varphi circ phi (a_i)=1) and begin{aligned} Vert aa_i-a_iaVert longrightarrow 0, Vert alpha.a_i-a_i.alpha Vert longrightarrow 0 (ain A, alpha in mathfrak {A}).