This paper will discuss the generalized Implicit Mapping Theorem for mappings . Only Y is required to be a Banach space. Z can be a Hausdorf topological vector space. For the existence and continuity, it sufices that X is a topological space and for diferentiability, X should be a topological vector space. Our proof is modelled on the classical proof given in [1]. With minor modifications, a definition of derivative for a mapping from a topological space to a topological group and the theorem holds when X is a topological space and Z is a Hausdorf topological group can be coined. In the final section, the Fundamental Theorem of Calculus for Henstock-Kurzeweil Integral for mappings assuming values in a Hausdorf topological vector space is obtained.

The abbreviation L.T. for a linear transformation is used and nbd for neighbourhood. The following facts about topological vector space from [2] are recalled. Topological Vector Space (TVS): A vector space over a field can be ) is called a topological vector space, abbreviated as TVS, if X is vector space and there is a topology on such that the addition as well as scalar multiplication, is continuous.

Remark 2.1.Any normed linear space is a TVS.

Definition 2.1. In a vector space V over , we define the following concepts.

For any set . A set A is called symmetric For any subset and is translate of A by the vector b and . A set A is called balanced if , for all scalars t with

Remark 2.2.Choosing shows that a balanced set is symmetric.

Example 1.Let A be a non-square plane rectangle. A is balanced as a subset of . But A is not balanced in the complex vector space C over C as

A set A is a TVS, is called bounded if for each nbd U of 0, if there exist 0 \end{document} ]]> such that , for all scalars t with . This is equivalent to any one of the following properties:

Clearly finite union of bounded sets is bounded and each finite set is bounded. It can be noted that translation and multiplication by a non-zero scalar are homeomorphisms and topology on a TVS can be defined using only neighbourhoods of 0. Also, it can be noted that each neighbourhood (nbd) of 0 contains a balanced closed nbd of 0. Scalar multiple or translate of a bounded set is bounded. Given a nbd U of 0, there exists a nbd V of 0 such that . Our vector spaces are over the field .

Definition 3.1.Let , be a TVS. A mapping is called diferentiable at a point x in if there exist a continuous L.T. such that,

uniformly for any h in a bounded set, that is, and given a nbd U of 0 in Y, there exist 0 \end{document} ]]> such that when We say is a δ-tangent or simply tangent.

Remark 3.1.The definition is clearly equivalent to the usual definition in Frechet spaces as shown in [3, 4] that the usual laws of diferentiation including the chain rule holds but a diferentiable mapping need not be continuous, however merely sequentially continuous.

Definition 3.2.A topological space X is called sequential if for any subset each cluster point a of A is the limit of a sequence of points in A. When X is a sequential space, each mapping diferentiable at a is continuous at a.

Definition 3.3 (Topology on is the space of continuous linear maps with the topology of uniform convergence on bounded subsets of E.

denotes a basic 0 − neighbourhood in the space . B is a bounded subset of X, V is a 0 − neighbourhood in Y.

Definition 3.4.A mapping is called a mapping in an open set E if the mapping is a sequentially continuous mapping , the space of continuous linear maps:

Definition 3.5(Norm of a linear mapping). In the case when and are normed spaces, the Norm of a continuous is sup . It is denoted by . It satisfies

Definition 3.6(Derivative for mappings of real variable). We note that when we consider mappings , where X is Hausdorf, the derivative can be identified with a vector in X and can be equivalently defined as in , we define derivative

Definition 3.7(Primitive of a mapping . Let be a closed cell in R. Let F and f be mappings defined on K with values in a Hausdorf TVS space . Let . A continuous mapping on K is said to be a primitive of f on K if F is rentiable on D, with

Definition 3.8.A division ofcell into mutually separated cells , for some is called a block in the partition of cell

We briefly discuss Henstock-Kurzweil Integral. The details can be found in [5, 6],[7, 8], [9], [10] and [11].

Definition 3.9 (Gauge). A gauge is a strictly positive function δ, defined on taking values in

A gauge allows the selection of tag points from an interval at which the Riemann sum is evaluated. It controls the size of the blocks in the partition. If and are two gauges on X, then there exist a gauge denoted by defined as , for all . Note:

Definition 3.10 (Tagged Partition of a cell K). By a tagged partition P of a cell , we mean a finite collection , where the collection is a partition of K, xk is a tag of on the real line). The is called as a tagged block.

Definition 3.11 (− fine partitions). For a gauge δ, a block of a tagged partition , is said to be , . The tagged partition P is called fine if each tagged block is . We say .

Definition 3.12 (Riemann sum). Let be a tagged partition for some gauge δ. We evaluate the mapping f at the tag points to form the Riemann sums , where is the length of the sub-interval .

The gauges form a directed set and implies . The Riemann sums form a net in the .

Definition 3.13.The mapping f with values in a TVS X is said to be Henstock–Kurzweil integrable over a cell , if there exist , such that , that is with the property, given a balanced nbd U of 0, a gauge such that, for

Theorem 4.1 (Implicit Function Theorem). Let be a topological space, a Banach space and Hausdorf TVS.

Part I: There exist a open nbd with , having the following property: there exist a continuous mapping , such that

Part II: When is TVS and the map is diferentiable with respect to x at the point with the derivative as , then is diferentiable at stands for composite of linear maps. The assertion holds in a nbd of a and when F is is

Proof. The proof works even if is a constant is replaced with which now satisfies the conditions in the theorem.

Theorem 5.1.The integral is well defined.

Proof. Cousin’s lemma [9, 11, 13, 14] ensures that given any gauge − fine tagged partition exists. Suppose I and I^′ are two values of the integral. Given a balanced nbd of 0 select a balanced nbd of 0 such that . As I is a value of the integral given , there exist a gauge such that . As I^′ is a value of the integral given , a gauge such that . Consider the gauge .

For we have , that is Since is Hausdorf, . If they should have disjoint nbds. So we arrive at a contradiction unless □

It is now shown that every derivative of a continuous function is integrable over a closed interval. This is not true for Riemann or Lebesgue integral.

Theorem 6.1 (Fundamental theorem of Calculus). Let be a primitive of a mapping f on a closed cell in R. Then

The proof of the theorem depends upon the following simple lemma. It must be noted that only consider balanced neighbourhoods of 0 have to be considered.

Lemma 6.2 (Straddle Lemma). If F is diferentiable at a point t, with the derivative denoted by , then for each nbd U of 0, there exist 0 \end{document} ]]> and a cell such that whenever x are in , that is , , that is, as .

Proof. As is diferentiable at the derivative being given by . As each nbd of origin contains a balanced nbd, given a balanced nbd of , consider a balanced nbd of 0 such that . So corresponding to , a number 0 \end{document} ]]> , such that . So, ]. Choose such that The result follows by the addition, using the triangle inequality on the real line and the order and . So . Hence we have , as □

Remark 6.1. [15]p and q need to straddle r, that is r is between p and q. The lemma states that the slope of the chord joining the points, with ordinates p and q and the slope of the tangent at the point whose ordinate is r are approximately equal as shown in Figure 1.

Proof. of theorem 6.1. We do this for every point t. So a gauge is obtained having the property. Let the gauge as in the Straddle lemma. Let be fine tagged partition of J. where, so that .

By Straddle lemma, , for each k adding finitely many terms. So, adding the constant mapping . □

Theorem 6.3 (Mean Value theorem for Vector valued mappings). [15]Let F be a mapping continuous on assuming values in a Hausdorf TVS X and diferentiable in , then , where

Proof. by the Fundamental Theorem of Calculus. By chain rule is a diferentiable mapping of θ and for , derivative of with respect to is the derivative of with respect to θ multiplied by . The proof follows from Fundamental Theorem of Calculus, noting that h is constant. □

Corollary 6.4. If is locally convex, that is each nbd contains a convex nbd then if is 0 in a path connected open set in , then is constant.

Proof. Since any nbd of a point a in the open set contains a convex nbd U so that any two points can be joined by a straight line segment By Mean Value Theorem, as the derivative is 0 we have ( b ) = F ( a )$ . So F is locally constant. As the domain is path connected, is constant. □

As in [16] and [17] one can deduce Mean Value theorem for a mapping 7 where both are TVS and X is locally convex.

In this article a new version of Mean Value theorem is obtained and proved on TVS using gauges. The Implicit Function Theorem is generalised on TVS. The conditions for existence, continuity and diferentiability are also provided for a mapping in TVS. In this article, TVS, Hausdorf TVS and Banach Space are linked with the mapping in generalised version of Implicit Function Theorem. It’s a fundamental tool in multivariable calculus and has applications in various fields, including physics, economics, and diferential geometry. This theorem allows us to express one or more variables in a system of equations as functions of the remaining variables, under certain conditions on the partial derivatives.