Let $1\leq p < \infty$ and let $(X,\Sigma, \mu)$ be a measure space. Let $F$ be the set of functions $f: X\to \mathbb R$ such that \(\vert \vert f\vert \vert _p = \left(\int_X\vert f\vert ^p\text d\mu\right)^{1/p}\ < \infty\) where $\vert \vert f\vert \vert _p$ is the p-norm. Equivalently, $F$ is the set of functions such that the $p$th power of their absolute value is integrable. Let $N = {f \in F \mid f = 0 \text{almost everywhere}}$. Because the integral of a positive function is zero if and only if it is zero almost everywhere, we have that $N$ is the kernel of the p-norm. The normed linear space given by the quotient vector space of $F$ by $N$ is $L_p(X,\mu)$.
For $p = \infty$, consider the set $G$ of functions $f:X \to \mathbb R$ for which there exists $c \in\mathbb R$ such that $\mu({\vert f\vert > c}) = 0$ and take the quotient vector space of this set by the equivalence relation $\sim$ on $G$ given by $f\sim g$ if $f(x) = g(x)$ almost everywhere on $X$. This space with the essential supremum of $f$ as the norm is $L_\infty(X,\mu)$.
Wikidata ID: Q305936