ON CERTAIN SUB-SPACE OF X

: The study of properties of space of entire functions of several complex variables was initiated by Kamthan [4] using the topological properties of the space. We have introduced in this paper the sub-space of space of entire functions of several complex variables which is studied by Kamthan.


INTRODUCTION:
The space of entire functions over the complex field C was introduced by V.G.Iyer [3] who defined a metric on this space by introducing a real-valued map on it.Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of topological bases etc. Kamthan [4] studied the properties of space of entire functions of several complex variables.Many of eminent mathematician such as Devendra [1], Hazem [2], Kumar [5], Mushtaq [6], Sirivastava [7], and others have contributed richly to space of entire functions of several complex variables.Let C denote the complex plane, and I be the set of non-negative integers.We write for , though my results can be easily extended to any positive integer n .Let therefore, X be the space of all entire functions C C f 2  : , where ) , ( , and assume that X is equipped with the topology T of uniform convergence on compact in n C .For more details see Kamthan [4].

1.
In this paper X denotes the space of all entire functions as in Kamthan [4]; S denotes the space of all double complex sequences,  I is the set of all positive integers, and  is a fixed element of S such that no coordinate element of  is zero , and is a linear sub-space of X .We now state two theorems, the proofs of which are left to the reader.Theorem 1.1.
if, and only if, Remark.The condition stated in Theorem 1.2 is not necessary.For, let  , be such that bounded sequences, so that, using Theorem 1.1, we note that both , so that is bounded.Proof.The sufficiency follows from Theorem 1.2 even without the extra hypothesis.To prove necessity with the extra hypothesis, suppose that Then this sequences has a subsequences, say Then, with the extra hypothesis, , and one of these tends to zero as This show that the condition stated in the theorem is necessary.

2.
) ( X is endowed with two topologies.One is the metric topology T inherited from X (vide [ Kamthan]   ), its metric d being It is known that X is a complete metric space under its usual topology (vide [4] ).We now prove that ) ), ( (   T X is complete under a condition to be stated in the following theorem.

Theorem 2.1.
) ), ( ( Let L be the infimum of the double sequence in the statement of the theorem.Then Thus each of the double sequences is a Cauchy sequence of complex number, so that each of these sequences tends to a limit as . Using this fact in (3) , we have in the second inequality in (4), we have . This leads us to the fact that .
We now show that , so that This proves that 0 lim , so that the condition is sufficient for ) ), ( ( , where , which is clearly not in X .Thus the Cauchy sequence , so that , in this case ) ), ( ( is in complete.Hence the condition is necessary.

3.
We have already seen that ) ( X can be endowed with two different topologies, viz., T and  T .We now state and prove a theorem relating to the comparability of T and  T .
Theorem 3.1.T is finer than  T if, and only if, be the supremum in the statement of the theorem.To prove that the above condition is sufficient, it is enough to . Since it is evidently linear , it is enough that we take , given an 0   , there is a

This
shows that , so that the condition is sufficient.ii) Necessity.Let the sequence in the statement of the theorem be unbounded .Then this sequence has a subsequence, say, , which is strictly increasing and tends to infinity.So the sequence Each of these polynomials is an element of . On the other hand, , so that the condition is necessary.

4.
Lastly we determine the form of a continuous linear functional on the complete metric space ) ), ( (   T X .Theorem 4.1.Every continuous linear functional  on the complete metric space ) ), ( ( , we can find an , , so that . It is clearly linear on ) ( X . We now show that it is continuous on ) ), ( ( . For this it is enough to show that , if    1 , , q p q p f is a double sequence in to be zero.We have already seen that f Before we proceed with the proof of this theorem, we shall state and prove a lemma.Lemma 4.2.A necessary and sufficient condition that the series Proof i) Sufficiency.Let (5) be bounded .Then we can find an 0 Conversely, let this sequence be bounded.Then by Lemma (4.2) ,