  Affine Bruhat--Tits buildings are geometric spaces extracting the
combinatorics of algebraic groups. The building of $\mathrm{PGL}$ parametrizes
flags of subspaces/lattices in or, equivalently, norms on a fixed
finite-dimensional vector space, up to homothety. It has first been studied by
Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The
space of seminorms compactifies the space of norms and admits a natural
surjective restriction map from the Berkovich analytification of projective
space that factors the natural tropicalization map. Inspired by Payne's result
that the analytification is the limit of all tropicalizations, we show that the
space of seminorms is the limit of all tropicalized linear embeddings
$\iota\colon\mathbb{P}^r\hookrightarrow\mathbb{P}^n$ and prove a faithful
tropicalization result for compactified linear spaces. The space of seminorms
is in fact the tropical linear space associated to the universal realizable
valuated matroid.
