The problem space is that area of problem statement and analysis. After all, if this is to be a solution there must have been a problem (or opportunity) that justified its creation. At least for me, I like to imagine this problem space as a line with inflections that represent the various needs, constraints and desires of the problem to be solved. During analysis, the line will not be distinct, but rather fuzzy and broad with a great deal of imprecision regarding what is really needed. Ideally this line (or surface if you want to think of it as 2 dimensional) is completely drawn to a specific level of detail before design begins but that is often not so.
What the designer attempts to do is create the ideal product that will match the requirements of the problem space with a product with capabilities complementary to the needs. The fitness for purpose will be the reflected in the gaps between the problem space and the solution space.
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