Stieltjes Limits in Continuous Function Spaces

The concept of limits in calculus was first discovered by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Both developed calculus with different but complementary approaches and notations. The Stieltjes limit is a generalization of the conventional limit, where the limiter is a function. This study defines the limit and Stieltjes limit on function-valued operators, which are bounded operators whose limiters are continuous functions. To support this definition, we first introduce the concepts of neighborhoods, continuous function operators, increasing operators, strictly increasing operators, decreasing operators, strictly decreasing operators, and their respective limits in function-valued operators. The results of this study reveal the necessary conditions and properties of limits and Stieltjes limits on function-valued operators, which share similarities with limits and Stieltjes limits in real-valued functions. These findings broaden our understanding of limit concepts in a more general context and provide new insights into operator analysis.