The set of real-valued continuous functions on R is vector space, V.
The integrable AND real-valued, continuous functions is a subspace, W of V.
(The "integrable functions" isn't a subspace, because integrable function might not be continuous!)
A different vector space is the real-valued, integrable functions, call it F.
The continuous, AND integrable, real-valued functions are subspace of F… is this set W again?
Best Answer
What is the difference between "integrable and real-valued and continuous" and "continuous and integrable and real-valued"? No matter how you permute these adjectives, they still define the same vector space, which injects as a sub-vector space into both $V$ and $F$. So the answer is yes, and $W$ is actually the intersection of $F$ and $V$, if we see them as sub-vector spaces of the vector space of all real-valued functions.