# References on Hausdorff Dimension on Non-Euclidean Vector Spaces?

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.

• Hausdorff dimension is defined for general metric spaces. en.m.wikipedia.org/wiki/Hausdorff_dimension Sep 21, 2023 at 3:52
• The vector space attribute is not particularly relevant, but instead being a metric space. See the references I cite in my answer to Metric dimensions properties not in $\mathbb{R}^d$. Also possibly relevant is my answer to Size of a function space, which was closed (and thus not viewable without sufficient reputation) but I reposted it here. Probably what you want to look at is Kolmogorov entropy of function spaces (has a huge literature). Sep 21, 2023 at 8:07