Editing Infinite-dimensional holomorphy (section)

==Vector-valued holomorphic functions defined in the complex plane==
A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-called ''vector-valued holomorphic functions'', which are still defined in the [[complex plane]] '''C''', but take values in a Banach space. Such functions are important, for example, in constructing the [[holomorphic functional calculus]] for [[bounded linear operator]]s.

<blockquote>'''Definition.''' A function ''f'' : ''U'' → ''X'', where ''U'' ⊂ '''C''' is an [[open subset]] and ''X'' is a complex Banach space, is called ''holomorphic'' if it is complex-differentiable; that is, for each point ''z'' ∈ ''U'' the following [[limit of a function|limit]] exists:

:<math>f'(z)=\lim_{\zeta\to z} \frac{f(\zeta)-f(z)}{\zeta - z}.</math></blockquote>

One may define the [[line integral]] of a vector-valued holomorphic function ''f'' : ''U'' → ''X'' along a [[rectifiable curve]] γ : [''a'', ''b''] → ''U'' in the same way as for complex-valued holomorphic functions, as the limit of sums of the form

:<math>\sum_{1 \le k \le n} f(\gamma(t_k)) ( \gamma(t_k) - \gamma(t_{k-1}) )</math>

where ''a'' = ''t''<sub>0</sub> < ''t''<sub>1</sub> < ... <  ''t''<sub>''n''</sub> = ''b''  is a subdivision of the interval [''a'', ''b''], as the lengths of the subdivision intervals approach zero.

It is a quick check that the [[Cauchy integral theorem]] also holds for vector-valued holomorphic functions. Indeed, if ''f'' : ''U'' → ''X'' is such a function and ''T'' : ''X'' → '''C''' a bounded linear functional, one can show that

:<math>T\left(\int_\gamma f(z)\,dz\right)=\int_\gamma (T\circ f)(z)\,dz.</math>
 
Moreover, the [[function composition|composition]] ''T'' <small>o</small> ''f'' : ''U'' → '''C''' is a complex-valued holomorphic function. Therefore, for γ a [[simple closed curve]] whose interior is contained in ''U'', the integral on the right is zero, by the classical Cauchy integral theorem. Then, since ''T'' is arbitrary, it follows from the [[Hahn–Banach theorem]] that

:<math>\int_\gamma f(z)\,dz=0</math>

which proves the Cauchy integral theorem in the vector-valued case.

Using this powerful tool one may then prove [[Cauchy's integral formula]], and, just like in the classical case, that any vector-valued holomorphic function is [[analytic function|analytic]].

A useful criterion for a function ''f'' : ''U'' → ''X'' to be holomorphic is that ''T'' <small>o</small> ''f'' : ''U'' → '''C''' is a holomorphic complex-valued function for every [[continuous linear functional]] ''T'' : ''X'' → '''C'''. Such an ''f'' is [[weak topology|weakly]] holomorphic.  It can be shown that a function defined on an open subset of the complex plane with values in a Fréchet space is holomorphic if, and only if, it is weakly holomorphic.