## Embedding problem of fuzzy number space. I.(English)Zbl 0757.46066

Summary: Using a theorem of R. Goetschel and W. Voxman [ibid. 18, 31- 43 (1986; Zbl 0626.26014)] we can embed fuzzy number space $$E^ 1$$ into a concrete Banach space $$\overline{C}[0,1]\times\overline{C}[0,1]$$. In addition, using a Rådström embedding theorem, M. L. Puri and D. A. Ralescu [J. Math. Anal. Appl. 91, 552-558 (1983; Zbl 0528.54009)] embed $$E^ 1$$ into a normed space $$X$$ with $$X=C-C$$. In fact, $$\overline{X}$$, the completion of $$X$$, is isometrically isomorphic to $$\overline{C}[0,1]\times\overline{C}[0,1]$$.

### MSC:

 46S40 Fuzzy functional analysis 03E72 Theory of fuzzy sets, etc. 54A40 Fuzzy topology

### Citations:

Zbl 0626.26014; Zbl 0528.54009
Full Text:

### References:

 [1] Bergstrom, H., Weak Convergence of Measures (1982), Academic Press: Academic Press New York · Zbl 0538.28003 [2] Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 31-43 (1986) · Zbl 0626.26014 [3] Natanson, N., Theory of Real Variables (1950), Soviet Academic Press: Soviet Academic Press Moscow, (in Russian) [4] Kaleva, O., Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301-317 (1987) · Zbl 0646.34019 [5] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35, 389-396 (1990) · Zbl 0696.34005 [6] Marłoka, M., On fuzzy integrals, (Albrycht, J.; Wisnieski, H., Proc. 2nd Polish Symp. on Interval and Fuzzy Mathematics (1987), Politechnika Poznansk: Politechnika Poznansk Poznan) [7] Puri, M. L.; Ralescu, D. A., Differentials for fuzzy functions, J. Math. Anal. Appl., 91, 552-558 (1983) · Zbl 0528.54009 [8] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl., 114, 409-422 (1986) · Zbl 0592.60004 [9] Rådström, H., An embedding theorem for spaces of convex sets, (Proc. Amer. Math. Soc., 3 (1952)), 165-169 · Zbl 0046.33304 [10] Yosida, K., Functional Analysis (1965), Springer-Verlag: Springer-Verlag Berlin · Zbl 0126.11504
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