Sentence examples for modular convergence implies from inspiring English sources

Without loss of generality, we assume L > 1 and ε < 1.Set β = ε h p.Since the modular convergence implies the norm convergence, so we can find δ > 0 suchthat I Φ ( y ) ≤ δ implies ∥ y ∥ Φ, p ≤ min { β 2, ε 1 p β 1 − 1 p 2 }.

Hence statistical convergence implies logarithmic statistical convergence.

The L1 convergence implies the a.e.

In certain cases (i.e, for special families of convex constraints), this weak convergence implies norm convergence.

Hence the uniform convergence implies that v is also continuous.

for all n > n 0, since weak convergence implies coordinatewise convergence.

It is well known that the complete moment convergence implies the complete convergence.

In view of the Borel-Cantelli lemma, complete convergence implies almost sure convergence.

Since the ordinary convergence implies ideal convergence, I is a regular sequential method [54].

Theorem 2.1 Statistical convergence implies logarithmic statistical convergence but converse need not be true.

Hence, from (2.7), one can clearly know that the complete moment convergence implies the complete convergence.