On subsets of Hilbert space having finite Hausdorff dimension

B. I. Shubov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 19, Tome 163 (1987), pp. 154-165

Résumé

Let $X_1$, $X_2$ be Hilbert spaces, $X_2\subset X_1$, $X_2$ is dense in $X_1$, the imbedding is compact, $M\subset X_2$, $\dim_H^{(i)}M$ and $h^{(i)}(M)$ are Hausdorff dimension and limit capacity (information dimension) of the set $M$ with respect to the metric of the space $X_i(i=1,2)$. Two examples are constructed. 1) An example of the set $M$ which is bounded in $X_2$ and such that a) $h^{(1)}(M)\infty$ (and therefore $\dim_H^{(1)}M\infty$) b) $M$ cannot be covered by a countable union of compact subsets of $X_2$ (and therefore $\dim_H^{(2)}M=\infty$)). 2) An example of the set $M$ which is compact in $X_2$ and such that $h^{(1)}(M)\infty$ and $h^{(2)}(M)=\infty$.