Editing Unbounded operator (section)

== Closed linear operators ==
{{Main|Closed linear operator}}

Closed linear operators are a class of [[linear operator]]s on [[Banach space]]s. They are more general than [[bounded operator]]s, and therefore not necessarily [[continuous function|continuous]], but they still retain nice enough properties that one can define the [[spectrum (functional analysis)|spectrum]] and  (with certain assumptions) [[functional calculus]] for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the [[derivative]] and a large class of [[differential operator]]s. 

Let {{math|''X'', ''Y''}} be two [[Banach space]]s. A [[linear transformation|linear operator]] {{math|''A'' : ''D''(''A'') ⊆ ''X'' → ''Y''}} is '''closed''' if for every [[sequence]] {{math|{''x''<sub>''n''</sub>} }} in {{math|''D''(''A'')}} [[limit of a sequence|converging]] to {{mvar|x}} in {{mvar|X}} such that {{math|''Ax<sub>n</sub>'' → ''y'' ∈ ''Y''}} as {{math|''n'' → ∞}} one has {{math|''x'' ∈ ''D''(''A'')}} and {{math|1=''Ax'' = ''y''}}. 
Equivalently, {{mvar|A}} is closed if its [[function graph|graph]] is [[closed set|closed]] in the [[direct sum of Banach spaces|direct sum]] {{math|''X'' ⊕ ''Y''}}.

Given a linear operator {{mvar|A}}, not necessarily closed, if the closure of its graph in {{math|''X'' ⊕ ''Y''}} happens to be the graph of some operator, that operator is called the '''closure''' of {{mvar|A}}, and we say that {{mvar|A}} is '''closable'''. Denote the closure of {{mvar|A}} by {{math|{{overline|''A''}}}}. It follows that {{mvar|A}} is the [[function (mathematics)|restriction]] of {{math|{{overline|''A''}}}} to {{math|''D''(''A'')}}.

A '''core''' (or '''essential domain''') of a closable operator is a [[subset]] {{mvar|C}} of {{math|''D''(''A'')}} such that the closure of the restriction of {{mvar|A}} to {{mvar|C}} is {{math|{{overline|''A''}}}}.

=== Example ===

Consider the [[derivative]] operator {{math|1=''A'' = {{sfrac|''d''|''dx''}}}} where {{math|1=''X'' = ''Y'' = ''C''([''a'', ''b''])}} is the Banach space of all [[continuous function]]s on an [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}}. 
If one takes its domain {{math|''D''(''A'')}} to be {{math|''C''<sup>1</sup>([''a'', ''b''])}}, then {{mvar|A}} is a closed operator which is not bounded.<ref>{{ harvnb | Kreyszig | 1978 | p = 294}}</ref> 
On the other hand if {{math|1=''D''(''A'') = [[smooth function{{!}}''C''<sup>∞</sup>([''a'', ''b''])]]}}, then {{mvar|A}} will no longer be closed, but it will be closable, with the closure being its extension defined on {{math|''C''<sup>1</sup>([''a'', ''b''])}}.