Let $V$ be the set of all positive real valued functions $f$ on $[a,b]$ such that $$\int_a^b f(x)\,\mathrm{d}x=2$$ with usual addition and scalar multiplication.
Then is $V$ be a vector space? If not, then give the properties that it doesn't follow.
This is a question which is given in my book. Now according to me, it is not a vector space as it is not closed under vector addition and neither is it closed under scalar multiplication, and also doesn't have a zero element.Since there is no zero element, the additive inverse property is also not followed.But the books says that additive inverse exists even though zero element does'nt exist.Is the book correct?