Could a subspace of a normed linear space be not a linear subspace? On page 38, Functional Analysis, Pater Lax: > Let $X$ be a normed linear space, $Y$ a subspace of $X$, The closure of $Y$ is a linear subspace of $X$. But on Wikipedia, linear subspace > A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces. I'm confused on the wording. What's the difference between subspace and linear subspace in Lax's book? Could a subspace of a normed linear space be not a linear subspace?

There is no difference: in functional analysis a subspace is intended as a linear subspace.

The only other subspaces you may have are topological subspaces: to avoid confusion you call them just subsets.