Given a fractal set in the Euclidean space $\mathbb{R}^n$, we can study it's Hausdorff dimension.

Common examples include:

(1) The middle-thirds Cantor set in $\mathbb{R}$.

(2) The Sierpinski Gasket in $\mathbb{R}^2$.

(3) The Menger Sponge in $\mathbb{R}^3$.

Most references I find (Kenneth Falconer's book, Wikipedia, etc.) usually discuss Hausdorff Measure and Dimension in the context of Euclidean spaces.

I am interested in studying fractal sets and their Hausdorff dimensions in other vector spaces. For example, how would one define Hausdorff dimension in a fruitful way over such spaces as:

(1) $C(X)$ - space of continuous functions over compact topological space $X$.

(2) $V^*(\mathbb{R})$ - dual space of all linear functionals $f:V\to \mathbb{R}$ where $V$ is some given vector space.

(3) $l^p$ - space of real-valued sequences $(x_k)_k$ such that $\Sigma_{k=1}^{\infty}|x_k|^p<\infty$.

These are just some examples that come immediately to mind.

I am mostly interested in references that pertain to $C(X)$ but anything related to Hausdorff dimension is good. Thank you.



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