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I am not sure what does the "topological" imply. Thanks.

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    $\begingroup$ The algebraic dual contains all functionals, even discontinuous ones. And the topological dual is again a Banach space under the dual norm. $\endgroup$ – Michael Greinecker Aug 7 '13 at 21:03
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The topological dual is defined as the space of all bounded linear functionals on your space. It is called this to differentiate it from the algebraic dual, which includes all linear functionals, including the non-bounded ones.

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  • $\begingroup$ Thanks. so the topology I give to the dual space will be the compact-open topology of maps from the Banach space to the scalar space. Or simply will be gotten by the norm defined as the sup of the norms of the image of the unit ball? $\endgroup$ – user89456 Aug 7 '13 at 21:10
  • $\begingroup$ It will just be the sup norm of the image on the unit ball. $\endgroup$ – Devlin Mallory Aug 7 '13 at 21:11

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