This book contains topological and distributional aspects of integral transforms with its applications. The topological and distributional concept of integral transforms for the solution of thermoelastic problems are the central theme of this book, find its roots and applications in branches of mathematics, aeronautical sciences and mechanical engineering which is helpful for students and researchers. We have constructed a new class or testing function space and using it other spaces and the countable union space have been constructed. We have also proved that is a complete locally convex, bornological Hausdorff topological t-space and obtained different important properties of above spaces. The mathematical approach of this book is to study new dimensions of thermoelastic problems in different areas.