The Problem Space is the problem and everything associated with the problem, including such things as the history of the problem; the stakeholders (those who benefit from the problem, those who contribute to the problem, and those who feel the problem most deeply, as pain).
The Solution Space, in contrast, constitutes the world of products, services, and policies that have been produced to address a particular problem.
Defining the problem space and writing a problem definition is the first step taken to solve a problem.
Venture Design, Design Thinking, Customer Discovery–these are different ways of talking about Customer Discovery and methods for navigating the problem space in order to develop solutions. In general, these approaches to entrepreneurship and lean development assume creators should set aside their own presumptions about best solutions and endeavor to listen to customers