I have to see that $L^2$ is a vector space, and would like to verify what I did. I excuse the long mail, but I am self-studying this, and is possible that I made wrong interpretations of what I am supposed to check.
I noticed that to check commutativity ($f + g = g + f$) or the existence of an additive inverse, I should know that if $f, g$ are square integrable, then the sum is also square integrable. And similarily, $-f$ is square integrable. So these two things would be a consequence of showing that, for $\alpha$ a scalar and $f,g$ square integrable, then $(\alpha f)(x) + g(x)$ is also square integrable, i.e $\int_a^b |[(\alpha f)(x) + g(x)]|^2 dx. $ By, https://www.physicsforums.com/threads/square-integrable-functions-as-a-vector-space.814979/ , I could prove that $$ \int_a^b |[(\alpha f)(x) + g(x)]|^2 dx \leq |\alpha|^2 \int^{a}_{b} | f(x) |^2 dx + \int^{a}_{b} | g(x) |^2 dx + 2\ | \overline{\alpha}| \sqrt{ \int^{a}_{b} |f(x)|^{2} dx \int^{a}_{b} |g(x)|^{2} dx},$$ where the first two terms are square integrable, and the third term... depends on two square integrable functions, but I'm not sure how I can say that it is a square int term. Are products of square ints square ints, and the square root of that is also square int?
Now, to check $L^2$ is a vector space, let $f,g,h$ square integrable functions in $[a,b]$:
$1) f + g $ is square int, i.e. $ \int_a^b |[(f + g)(x)]|^2 dx $ = $ \int_a^b |[(f)(x) + g(x)]|^2 dx$ = $ \int_a^b |[g(x) + f(x)]|^2 dx$ = $ \int_a^b |[(g + f)(x)]|^2 dx ,$ by definition of sum of functions and as they are evaluated in a field, the sum is commutative.
2) Because of associativity in the field $ \int_a^b |[(f + g)(x)+ h(x)]|^2 dx $ = $ \int_a^b |[f(x) + (g+ h)(x)]|^2 dx $ so, $(f + g) + h = f + (g +h)$ is in $L^2 .$
3) If $n(x) = 0$ is the additive identity, then $ \int_a^b |[f(x)+ n(x)]|^2 dx $ = $ \int_a^b |[f(x)+ 0]|^2 dx $ = $ \int_a^b |[f(x)]|^2 dx $ for every $f \in L^2.$
4) If $-f(x)$ is the additive inverse for $f$, then $f + (-f)$ is square integrable, and $ \int_a^b |[f(x) -f(x)]|^2 dx $ = $ \int_a^b |0| dx $ = $0$ for every $f \in L^2.$
5) $ \int_a^b | \alpha( \beta f(x))|^2 dx $ = $ ( \int_a^b |(\alpha\beta)f(x)|^2 dx), $ by properties of the field .
6) If $g(x) =1$ is the multiplicative identity, then $ \int_a^b |g(x) f(x)|^2 dx $ = $ \int_a^b |1 f(x)|^2 dx $ = $ \int_a^b |f(x)|^2 dx $ for every $f \in L^2.$
7) If $(f + h) (x)$ is square int, them $ \int_a^b |\beta (f + h) (x)|^2 dx = \int_a^b |(\beta f)(x) + (\beta h)(x)|^2 dx ,$ so equality holds and $(\beta f)(x) + (\beta h)(x)$is square integrable.
8) As $f$ is square integrable, then $(\alpha + \beta)f$ is square integrable, e.g. $ \int_a^b |[(\alpha + \beta )f] (x)|^2 dx$ = $ \int_a^b |\alpha f (x) + \beta f (x)|^2 dx,$ by field properties. So equality holds and $\alpha f (x) + \beta f (x)$ is square integrable.