I am following the proof of Riesz Representation Thm, where $\textbf{"subordinate partition of unity"}$ is used. However, I have problem in understanding the definition.
Could there be any (possibly better geometric) elaboration on the definition (Especially if we assumed below $X=\mathcal{C}[X]$ ) .
This is the def. I read;
$\textbf{Definition}$: Let $\{U_{\alpha}\}_{\alpha \in I}$ be an open cover of a metric space X. A partition of unity subordinates to $\{U_{\alpha}\}_{\alpha \in I}$, is $\{\Psi_{U_{\alpha}}\}_{\alpha \in I}$ , such that
$$\Psi_{U_{\alpha}} \in \mathcal{C}(X) ,\ 0\leq \Psi_{U_{\alpha}} \leq \mathcal{X}_{[U_{\alpha}]} \ \text{and} \ \sum_{\alpha\in I} \Psi_{U_{\alpha}}(x) = 1$$
Best Answer
All this means is that for each $\alpha\in I,$ the support of $\Psi_{U_{\alpha}}$ is contained in $U_{\alpha},\ $ the values of each $\Psi_{U_{\alpha}}$ lie in the interval $[0,1]$ and for each $x\in X$, the sum $\sum_{\alpha\in I} \Psi_{U_{\alpha}}$ is actually finite and equal to $1$.
The geometric idea is that you have a bunch of functions, each one of which is concentrated in a covering element of the metric space $X$. This allows you, in many cases, to move from the local to the global: if you have for example, smooth functions defined only in a neighborhood of a given point, by using the partition of unity, you can cobble them together to get a globally defined function that retains the local property.