Notational difference between "+" and "∪" for defining a vector space from Lebesgue spaces.

Question

I have a space $X$ defined in the literature as,

$X \in L^{\frac{3}{2}} + L^{\infty}$

is this denoting the union of the $L^{\frac{3}{2}}$ and $L^{\infty}$ spaces? If so, could it equally be written as,

$X \in L^{\frac{3}{2}} \cup L^{\infty}$

or is there a deeper difference between these expressions that I'm missing?

Answer 1

This is not the same. The expression

$$L^{\frac{2}{3}}+L^\infty=\{x+y|x\in L^{\frac{2}{3}},y\in L^\infty\}$$

is the sum of vector spaces which is different from the union of both spaces.

See e.g. the Wiki-page about sum of subspaces and this paper regarding sum of Lebesgue spaces.