Abstract
Let X X and Y Y be two Banach spaces. We show that a bounded bilinear form m : X × Y → C m:X \times Y \to {\mathbf {C}} is Arens regular iff it is weakly compact. This result permits us to find very short proofs of some known results as well as some new results. Some of them are: Any C ∗ {C^*} -algebra, the disk algebra and the Hardy class H ∞ {H^\infty } are Arens regular under every possible product. We also characterize the Arens regularity of certain bilinear mappings.
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