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So I was told that if $p,q in mathbb{L}^2 _mathbb{C} [a,b]$ which is the set of all square integrable functions on the interval $[a,b]$ then:
$$(p,q) = int^b_a{dx p^*(x)q(x)}$$
is a scalar product.
However, for this to be the case $(p,p) = 0 leftrightarrow p =0$ should be satisfied. $(0,0) = 0$ is obvious but on $mathbb{L}^2 _mathbb{C}$ there can be other functions that can yield 0 upon integration (e.g. a function that takes a finite value at one point but is zero everywhere else), so how can this be true?

You'll need some measure theory for this. $L^2$ actually isn't a space of functions but a space of equivalence classes of functions. Two square-integral functions $f$, $g$ are defined to be equivalent if they differ only on a set of measure zero (If you don't know measure theory, think of this as a set with zero length/volume/area).

The function in your example differs from zero in one point only and since the measure of a one-point set is zero, your function is actually identical to zero in $L^2$.

More generally, if a function $p$ is nonzero on a set $A$ of positive measure, then $f = |p|^2 = p^* p$ is also nonzero on $A$. If you divide $A$ as to $A = bigcup_{n geq 1} A_n = bigcup_{n geq 1} {x in A | f(x) > frac1n}$ you get that one of the $A_n$ must have nonzero measure and therefore $int_a^b f(x) ,mathrm{d}x > frac1n mu(A_n) > 0$. (here $mu$ denotes the measure). But this means that a function $p$ which is not equivalent to $0$ in $L^2$ also has a positive "scalar square" $(p, p)$, proving definiteness of the scalar product on $L^2$.

The set $mathbb{L}^2[a, b]$ is defined as the quotient space $mathcal{L}^2[a, b]$ / $mathcal{N}$. Thus $mathbb{L}^2$ is actually a set of equivalence classes. Here $mathcal{N}$ denotes the set of all functions that are non-zero on a set with Lebesgue measure zero. $mathcal{L}^2[a, b]$ denotes the set of all square-integrable functions (i.e. all functions f such that $||f||_2^2 = langle f, frangle = int_a^b |f|^2(x) dx < infty$.) So $mathbb{L}^2$ is the set of all equivalence classes of functions that are equal almost everywhere (i.e. they differ only on a set with measure $0$).

Note that the above 2-norm on $mathcal{L}^2[a, b]$ isn't an actual norm, as $||f||_2^2 = 0 iff f equiv 0$ is obviously not satisfied, similarly as how $langle f, f rangle = 0 iff f equiv 0$ is not satisfied on $mathcal{L}^2[a, b]$. However, writing $f = 0$ for an $f in mathbb{L}^2$ means that $f$ is the equivalence class of the functions $f in mathcal{L}^2$ that are $0$ almost everywhere.

Note that, $||f||_2^2 = 0 iff f = 0$ and $langle f, f rangle = 0 iff f = 0$ are satisfied on $mathbb{L}^2$ (contrary to $mathcal{L}^2$), since the Integral of $|f|^2$ is $0$ if and only if $f$ is zero almost everywhere (i.e. if $f$ is the equivalence class of functions that are zero almost everywhere).

EDIT: Some further reading (also for more general $mathbb{L}^2$ spaces than just $mathbb{L}^2[a,b]$): https://en.wikipedia.org/wiki/Lp_space#Lp_spaces