I am a student learning Iwasawa theory. I am so sorry if this post is too trivial for this site. I posted it on math.stackexchange yesterday but obtained no responce.

A quite basic object is the Iwasawa algebra. A basic version is as follows:

Let $L/\mathbb{Q}_p$ be a finite extension (i.e. a local number field) where $p$ is a prime (if necessary, we may assume $p \geq 3$ or $p \geq 5$.) Let $\mathcal{O}_L$ be its ring of integer. Then we define $\Lambda := \mathcal{O}_L[[T]]$ as the ring of formal power series of one indeterminate $T$ with $\mathcal{O}_L$-coefficient, called an **Iwasawa algebra**.

I have see in many papers that people regard $\Lambda$ as a **disk**, or regard the "**rational Iwasawa algebra**" $\Lambda_{\mathbb{Q}}:= \Lambda \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$ (I am still wondering if this is correct, as a tensor product over $\mathbb{Z}_p$?) as a disk. Moreover, people often say that some kind of a $p$-adic L-function $\mathcal{L}_p$ is in the disk, and **"a small neighborhood"** of the $\mathcal{L}_p$ satisfies certain properties.

**MY QUESTION**: How to make such statements precise? In other words, how can people view the Iwasawa algebra as a disk, talk about its elements as points and the neighboorhood of the points.

Moreover, quite recently, there is a notion called **"shrinked Iwasawa algebra"** $\Lambda_{m}:= \mathcal{O}_L[[p^{-m}T]] \subset \Lambda$. The author stated that this can be regarded as shrinking the disk into smaller radius, just as the name shows.

**My further question**: How do people regard this "shrinked Iwasawa algebra" as a disk of smaller radius?

Put a step further, in more applications, we may consider $\mathbb{Z}_p$-extensions $\mathcal{K}_{\infty}/\mathcal{K}$ of Galois group $\mathbb{Z}_p^d$ for $d > 1$. Then the corresponding Iwasawa algebra should be $\Lambda_{(d)} := \mathcal{O}_L[[T_1, \ldots, T_d]]$. Then

**Can we also regard $\Lambda_{(d)}$ as some kind of disk** and further generalize the "shrinked Iwasawa algebra"?

Thank you all for commenting and answering! Any references on explaining these are also welcome!