What's the definition of proper subspace of a vector space used in Rudin's Functional analysis

Question

I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology.

I'm reading through chapter 1 again, and through the glossary as well but I cannot find the definition used.

I guess there must be a standard definition then,

What is such definition?

Answer 1

$V$ is a proper subspace of $X$ if $V$ is a subspace of $X$ and $V\subsetneq X$.

I would guess that there is no definition of proper subspace in the book, since a proper subspace is a subspace that is also a proper subset.