Closed range theorem

The rate of completed image is a mathematical sentence from the branch of functional analysis. He makes a statement about when the image of a continuous linear operator is complete.

Motivation

Is a continuous linear operator between normed spaces, as explained by one of the dual operator.

For a subspace is, that is the subspace in the dual space, which consists of all continuous linear functionals which vanish on. For a subspace is defined analogously a subspace in the formula. ( In the literature one finds for the designation and thus takes an ambiguity in the name of purchase.)

Using the separation theorem (or the set of Hahn- Banach ) shows you and, where ker and stand in for the core and image of an operator. Such a relationship is familiar from linear algebra. Accordingly, one would expect a similar formula as expected, but in general can not be considered, because is always complete, however, the image of a continuous linear operator generally not. If, for example, the Banach space of all null sequences, so is a continuous linear operator with dense (ie not - completed ) image. Such phenomenon may in linear algebra, that not occur in finite spaces. In order to reach the expected from linear algebra formula, so you must have the seclusion of the image space provide. This proves to be sufficient and equivalent to the corresponding statement about the dual operator:

Set by the completed image

Let and be Banach spaces and a continuous linear operator. Then the following statements are equivalent:

• Is completed.
• .
• Is completed.
• .

For general normed spaces this theorem does not apply. So, for example, has a self- image (because is surjective ), but the dual operator, which is the usual identifications in subsequent spaces equal to the inclusion mapping, has no closed image.

Application

Are and continuous linear operators between Banach spaces, so you can from the sequence

Form, where 0 is the zero vector space stand, and raise the question of the accuracy. The sequence is specified if and exactly when the dual sequence

Is exact. Indeed, if the output sequence exactly, so the pictures are from and completed with. Therefore, by the above theorem are also the images of and complete, and it follows

This means exactness of the dual sequence. Just follow the accuracy of the output sequence from the exactness of the dual sequence.