The Theory of Vector-Valued Function in Locally Convex Space
Lixin Ma

Abstract

In this paper, the vector-valued regular functions are extended to the locally convex space. The residues theory of the functions in the locally convex space is achieved. Thereby the Cauchy theory and Cauchy integral formula are extended to the locally convex space.

Share and Cite:

Ma, L. (2012) The Theory of Vector-Valued Function in Locally Convex Space. Applied Mathematics, 3, 903-906. doi: 10.4236/am.2012.38133.

1. Introduction

The properties of analytic functions have been given in references [1,2]. The theory of analytic functions was extended to vector valued function in reference [3].

In this paper, we extended the theory of vector valued function to locally convex space.

Let be a complete Hausdorff locally convex space on the real or complex domain, and be the sufficient directed set of semi norms which generates the topology of. We denote the ad joint space of by, i.e. is the set of linear bounded functions on.

Definition 1 Let be a vector function defined on a domain with values in. If there is an element such that the difference quotient

tends weakly(strongly) to as

, we call the weakly (strongly) derivative of at. We also say that is weakly (strongly) derivative at in. We call weakly (strongly) derivative in.

Definition 2 A vector function is

1) weakly continuous at if

for each.

2) strongly continuous at if

for each.

Definition 3 A vector function is said to be regular in if is regular for every, where range of is in. If a vector valued function is regular in, then is called an entire function or said to be entire.

Theorem 1 [4] (Cauchy) If is a regular vector-valued function on the domain with values in the locally convex space. Let be a closed path in, and assume that is homologous to zero in, then

where c is a circle.

Proof For any linear bounded functional, we have

Hence

Theorem 2 [5] (Cauchy integral formula) Let be a regular vector-valued function on the domain with values in the locally convex space. Let be a closed path in, and assume that is homologous to zero in, and let be in and not on. Then

(1)

where is the index of the point with respect to the curve.

Proof For any linear bounded functional, we have

.

Then

2. The Main Conclusions

Theorem 3 Given the power series

. (2)

Set. Then the power series (2)

is absolutely convergent for and divergent for. The power series (2) convergence to a regular function on with values in, the convergence being uniform in every circle of radius less than.

Proof First, we will prove the power series (2) is absolutely convergent for and divergent for.

By Theorem 1, for any, we have

where.

Let, then

where. Thus the power series (2) is absolutely convergence. But for, if we suppose the power series (2) is convergence, it is contradict with the radius is. So the power series (2) is absolutely convergent for and divergent for.

Secondly, for any linear bounded functional, we have

.

The right side series convergence to a regular function on with values in. So is regular in the circle and the convergence being uniform.

Definition 4 Let have an isolated singularity at and let

(3)

where

(4)

be its Laurent Expansions about. The residue of at is the coefficient. Denote this by.

Theorem 4 Let be a regular vector-valued function except for a finite number of points in the domain. Let be a closed path in, and assume that is homologous to zero in, and let be in and not on. Then

(5)

Proof For any linear bounded functional, we have

.

Then

Theorem 5

1) If has a pole of order one at a point

then

(6)

2) If has a pole of order at a point then

(7)

Proof Because has a pole of order at a point, then can be written in the form

where is regular and nonzero at.

So has a power series representation

in some neighborhood of. It follows that

in some neighborhood of. Then we have formula (7)

Obviously, when, the formula (7) is formula (6).

Theorem 6 If

where for and if exists, then exist and has a pole with order at.

Proof Since

For any linear bounded functional, we have

aswhere is sufficiently small. Thus

.

It follows that

Therefore

where

Remark: exist, this condition is important.

For example, in, we define, where and For any linear bounded functional

.

Thus is a B-algebra, and. We set

where and. It follows that is zero with order one, but

With order three.

Theorem 7 If and are regular in with values in and if, , the points having a limit point in, then in.

Proof For any linear bounded functional, we have

So

.

Theorem 8 Let be defined in a domain of the extended plane and on its boundary, regular in and strongly continuous in. If

then either or in.

Proof For any linear bounded functional, we have

.

But except is constant,. So either or in.

Remark: Unlike the classical case, may have a minimum other zero in as the following example shows.

For example, Let be a Banach space of complex pairs, , where.

Set

Then

, for

and

for.

Theorem 9 If is regular in, and if is bounded in, then constant element.

Proof For any linear bounded functional, we have

.

So is bounded in, then is constant.

Suppose is not constant, then exist two point such that

.

Thus exist satisfy

.

This is contradict with is constant. So constant element.

Theorem 10 If is regular in the unit circle, satisfy the condition and. Then

.

Proof For any linear bounded functional, we have

.

Since every point, their exist a bounded function such that

.

So

.

Then

.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] L. V. Ahlfors, “Complex Analysis,” McGraw-Hill, New York, 2004. [2] S. Lang, “Complex Analysis,” Springer-Verlag, New York, 2003. [3] C. G. Hu, “Vector-Valued Function and Its Applications,” Kluwer Academic Publishers, Dordrecht, Boston, London, 1992. [4] C. P. Xie and Y. S. Zhao, “Basic Boundary Value Problem for Complex Harmonic Functions,” Beijing Math, 1997, Vol. 3, No. 1, pp. 20-35. [5] Z. Zhen, “Schwarz’s Problems for Some Complex Partial Differential Equations of Second Order,” Beijing Math, 1996, Vol. 2, No. 1, pp. 131-137.