Abstract
A strong law of large numbers is proved for tight, independent random elements (in a separable normed linear space) which have uniformly bounded $p$th moments $(p > 1)$. In addition, a weak law of large numbers is obtained for tight random elements with uniformly bounded $p$th moments $(p > 1)$ where convergence in probability for the separable normed linear space holds if and only if convergence in probability for the weak linear topology holds.
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