Suppose we have a bounded metric space $(X,d)$ and a countable measurable partition $Q$ of $X$ (with respect to some probability measure $\mu$) such that $\mu(q)>0$ for all $q\in Q$. We know that the Holder space with exponent $\alpha$, denoted $C^\alpha(X)$, consisting of those functions $f:X\to \mathbb{R}$ with $\|f\|_\alpha<\infty$, where $\|f\|_\alpha=\|f\|_\infty +|f|_\alpha$ (here $\|f\|_\infty=\sup\limits_{x}|f(x)|$ and $|f|_\alpha=\sup\limits_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\alpha}$ ) is a Banach space, whose unit ball is precompact in $\mathcal{L}^\infty(Y)$.
We define $\mathcal{C}$ to consist of those functions $f:X\to \mathbb{R}$ for which $\|f\|_C=\sum\limits_{q\in Q}\left\|f|_q\right\|_\alpha\mu(q)<\infty$. Similarly, we define $\mathcal{L}$ to consist of those functions $f:X\to \mathbb{R}$ for which $\|f\|_L=\sum\limits_{q\in Q}\left\|f|_q\right\|_\infty\mu(q)<\infty$. These can be shown to be Banach spaces.
My question is, can we show that the unit ball of $\mathcal{C}$ is compact in $\mathcal{L}$?
I have tried (and failed) for a while to show this. For example, one method I tried was the following: Take a subsequence $(f_n)\subset \mathcal{C}$ with $\|f_n\|_C\leq 1$. Writing $P$ as $\{p_1,p_2,\ldots\}$, using precompactness of the unit ball of $C^\alpha(Y)$ in $\mathcal{L}^\infty(Y)$, and using a diagonal argument, I constructed a subsequence $(g_n)$ such that $g_n|_{p_i}\to f_{p_i}$ as $n\to\infty$ for some $f_{p_i}\in\mathcal{L}^\infty(Y)$ and all $i\in \mathbb{N}$. Setting $f=\sum\limits_{i=1}^\infty f|_{p_i}$, I want to show that $g_n\to f$ in $\mathcal{L}$. However, I am not too sure how to show this.
Note that $\|g_n-f\|_\mathcal{L}=\sum\limits_{i=1}^\infty\left\|g_n|_{p_i}-f|_{p_i}\right\|_\infty\mu_Y(p_i)$. Moreover, $f\in \mathcal{L}$ by construction (with norm $\leq 1$), so that $\|g_n-f\|_\mathcal{L}\leq 2$.
Can we deduce the required convergence from this information?