In
mathematics, Kōmura's theorem is a result on the
differentiability
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
of
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
-valued functions, and is a substantial generalization of Lebesgue's theorem on the differentiability of the
indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
, which is that Φ :
, ''T''nbsp;→ R given by
:
is differentiable at ''t'' for
almost every 0 < ''t'' < ''T'' when ''φ'' :
, ''T''nbsp;→ R lies in the
''L''''p'' space ''L''
1(
, ''T'' R).
Statement
Let (''X'', , , , , ) be a
reflexive Banach space and let ''φ'' :
, ''T''nbsp;→ ''X'' be absolutely continuous. Then ''φ'' is (strongly) differentiable almost everywhere, the derivative ''φ''′ lies in the
Bochner space ''L''
1(
, ''T'' ''X''), and, for all 0 ≤ ''t'' ≤ ''T'',
:
References
* (Theorem III.1.7)
{{DEFAULTSORT:Komuras theorem
Theorems in measure theory
Theorems in functional analysis