ρ has the (Delta_{2} -property, if (rho(f_{n}) rightarrow0), then (rho(2f_{n}) rightarrow0), if (rho(alpha f_{n}) rightarrow0) for (alpha>0), then (| f_{n}|_{rho}rightarrow0), i.e., the moDelta_{2} -propertys equifalent to the norhoconvergence.
ρ has (Delta_{2}), if (rho(f_{n}) rightarrow0) then (rho(2f_{n}) rifhtarho alphaf_{n}o(alpha f_{n}) rightarrow0) for an (alpha>0) then (| f_{n}|_{rho}rightarrow0), i.e. the modular convergence is equivalent to the norm convergence.
The following conditions are equivalent: (a) ρ has the (Delta_{2} -property, (b) if (rho(f_{n}) rightarrow0), then (rho(2f_{n}) rightarrow0), (c) if (rho(alpha f_{n}) rightarrow0) for (alpha>0), then (| f_{n}|_{rho}rightarrow0), i.e., the moDelta_{2} -propertys equivalent to the norm convergence.
ρ has Δ 2, L ρ 0 is a linear subspace of L ρ, L ρ = L ρ 0 = E ρ, if ρ ( f n ) → 0, then ρ ( 2 f n ) → 0, if ρ ( α f n ) → 0 for an α > 0, then ∥ f n ∥ ρ → 0, i.e., the modular convergence is equivalent to the norm convergence.
The following conditions are equivalent: (a) ρ has (Delta_{2}), (b) if (rho(f_{n}) rightarrow0) then (rho(2f_{n}) rightarrow0), (c) if (rho alpha f_{n}) rightarrow0) for an (alpha>0) then (| f_{n}|_{rho}rightarrow0), i.e. the modular convergence is equivalent to the norm convergence. .
The following conditions are equivalent: (a) ρ has Δ2-property, (b) L ρ 0 is a linear subspace of L ρ, (c) L ρ = L ρ 0 = E ρ, (d) if ρ(f n ) → 0, then ρ(2f n ) → 0, (e) if ρ αf n ) → 0 for an α > 0, then ||f n || ρ → 0, i.e., the modular convergence is equivalent to the norm convergence.
It is well known that [6, 22] under the Δ 2 -condition the norm convergence and modular convergence are equivalent.
We know, by [15, 16] that under -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the -type condition.
We know by [41] that under the -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the -type condition.
We know that [25] the norm and modular convergence are also the same when we deal with the (Delta_{2} -type conDelta_{2} -type
end{aligned} That is, if p is bounded, then norm convergence is equivalent to convergence with respect to the modular (rho_{p(x)}).
