Editing Dual space (section)

=== Examples ===

Let 1 < ''p'' < ∞ be a real number and consider the Banach space ''[[Lp space#The p-norm in countably infinite dimensions|ℓ<sup>&thinsp;p</sup>]]'' of all [[sequence]]s {{math|1='''a''' = (''a''<sub>''n''</sub>)}} for which

:<math>\|\mathbf{a}\|_p = \left ( \sum_{n=0}^\infty |a_n|^p \right) ^{\frac{1}{p}} < \infty.</math>

Define the number ''q'' by {{math|1=1/''p'' + 1/''q'' = 1}}. Then the continuous dual of ''ℓ''<sup>&thinsp;''p''</sup> is naturally identified with ''ℓ''<sup>&thinsp;''q''</sup>: given an element <math>\varphi \in (\ell^p)'</math>, the corresponding element of {{math|''ℓ''<sup>&thinsp;''q''</sup>}} is the sequence <math>(\varphi(\mathbf {e}_n))</math> where '''<math>\mathbf {e}_n</math>''' denotes the sequence whose {{mvar|n}}-th term is 1 and all others are zero. Conversely, given an element {{math|1='''a''' = (''a''<sub>''n''</sub>) ∈ ''ℓ''<sup>&thinsp;''q''</sup>}}, the corresponding continuous linear functional ''<math>\varphi</math>'' on {{math|''ℓ''<sup>&thinsp;''p''</sup>}} is defined by

:<math>\varphi (\mathbf{b}) = \sum_n a_n b_n</math>

for all {{math|1='''b''' = (''b<sub>n</sub>'') ∈ ''ℓ''<sup>&thinsp;''p''</sup>}} (see [[Hölder's inequality]]).

In a similar manner, the continuous dual of {{math|''ℓ''<sup>&thinsp;1</sup>}} is naturally identified with {{math|''ℓ''<sup>&thinsp;∞</sup>}} (the space of bounded sequences).
Furthermore, the continuous duals of the Banach spaces ''c'' (consisting of all [[limit of a sequence|convergent]] sequences, with the [[supremum norm]]) and ''c''<sub>0</sub> (the sequences converging to zero) are both naturally identified with {{math|''ℓ''<sup>&thinsp;1</sup>}}.

By the [[Riesz representation theorem]], the continuous dual of a Hilbert space is again a Hilbert space which is [[antiisomorphic|anti-isomorphic]] to the original space.
This gives rise to the [[bra–ket notation]] used by physicists in the mathematical formulation of [[quantum mechanics]].

By the [[Riesz–Markov–Kakutani representation theorem]], the continuous dual of certain spaces of continuous functions can be described using measures.