Editing Banach space (section)

====Bidual====
{{See also|Bidual|Reflexive space|Semi-reflexive space}}

If <math>X</math> is a normed space, the (continuous) dual <math>X''</math> of the dual <math>X'</math> is called the '''{{visible anchor|bidual}}''' or '''{{visible anchor|second dual}}''' of <math>X.</math> 
For every normed space <math>X,</math> there is a natural map,
<math display="block>\begin{cases}
F_X\colon X \to X'' \\
F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X'
\end{cases}</math>

This defines <math>F_X(x)</math> as a continuous linear functional on <math>X',</math> that is, an element of <math>X''.</math> The map <math>F_X \colon x \to F_X(x)</math> is a linear map from <math>X</math> to <math>X''.</math> 
As a consequence of the existence of a [[Banach space#Dual space|norming functional]] <math>f</math> for every <math>x \in X,</math> this map <math>F_X</math> is isometric, thus [[injective]].

For example, the dual of <math>X = c_0</math> is identified with <math>\ell^1,</math> and the dual of <math>\ell^1</math> is identified with <math>\ell^{\infty},</math> the space of bounded scalar sequences. 
Under these identifications, <math>F_X</math> is the inclusion map from <math>c_0</math> to <math>\ell^{\infty}.</math> It is indeed isometric, but not onto.

If <math>F_X</math> is [[surjective]], then the normed space <math>X</math> is called ''reflexive'' (see [[Banach space#Reflexivity|below]]). 
Being the dual of a normed space, the bidual <math>X''</math> is complete, therefore, every reflexive normed space is a Banach space.

Using the isometric embedding <math>F_X,</math> it is customary to consider a normed space <math>X</math> as a subset of its bidual. 
When <math>X</math> is a Banach space, it is viewed as a closed linear subspace of <math>X''.</math> If <math>X</math> is not reflexive, the unit ball of <math>X</math> is a proper subset of the unit ball of <math>X''.</math> 
The [[Goldstine theorem]] states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. 
In other words, for every <math>x''</math> in the bidual, there exists a [[Net (mathematics)|net]] <math>(x_i)_{i \in I}</math> in <math>X</math> so that
<math display="block>\sup_{i \in I} \|x_i\| \leq \|x''\|, \ \ x''(f) = \lim_i f(x_i), \quad f \in X'.</math>

The net may be replaced by a weakly*-convergent sequence when the dual <math>X'</math> is separable. 
On the other hand, elements of the bidual of <math>\ell^1</math> that are not in <math>\ell^1</math> cannot be weak*-limit of {{em|sequences}} in <math>\ell^1,</math> since <math>\ell^1</math> is [[#Weak convergences of sequences|weakly sequentially complete]].