**6. Conclusions**

In this paper, we discussed the preimage problem in the space where the relationship between objects is determined by closeness. We showed that any metric can be transformed into closeness and therefore, the latter is weaker than the former. We interlaced closeness with fuzzy partition and characterized both by the closeness matrix.

We expressed similarity between functions based on their images (coinciding with vectors of scaled F-transform components) computed using the closeness matrix. By that (and by setting the basic structure of the space without metric, or norm), we contributed to the mathematical theory in the field of functional analysis. We formulated the preimage problem using the language of matrix calculus. The preimage problem solution is given by (i) a weighted arithmetic mean, (ii) any right inverse of the closeness matrix or (iii) any element of a certain affine subspace. Singular value decomposition was applied to describe the problem and its solution.

We defined the notion of compatible set of basic functions and found conditions under which the inverse F-transform with respect to the given and compatible set of basic functions forms a solution to the preimage problem.

Theoretical results were illustrated by numerical examples. They demonstrate, e.g., that requiring reflexivity of closeness can be counterproductive.

The future research will be focused on imposing further conditions on both collections of basic functions (*At*'s and *Bt*'s) to reveal stronger connections between the spaces *A* and *B*.

**Author Contributions:** Investigation, J.J. and I.P.; resources, J.J. and I.P.; writing—original draft preparation, J.J.; writing—review and editing, J.J. and I.P.; supervision, I.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** The support of the project SGS20/PˇrF-MF/2022 of the University of Ostrava is greatly appreciated.

**Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Symbols and Abbreviations**

The following symbols and abbreviations are used in this manuscript:

