unconditional convergenceの例文

• For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence.
• Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general, even if the space is Banach, although the implication holds in \ mathbb { R } ^ d.
• Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general, even if the space is Banach, although the implication holds in \ mathbb { R } ^ d.
• Since a series with values in a finite-dimensional normed space is absolutely convergent if each of its one-dimensional projections is absolutely convergent, it follows that absolute and unconditional convergence coincide for "'R " "'n "-valued series.
• When " X " & thinsp; is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if " X " & thinsp; is a Banach space, this is equivalent to say that for every sequence of signs " ? " " n " = ?, the series