The following is an excerpt of a note in topological vector spaces.
I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we usually call "seminorm". Does anybody see the term "semi-pseudonorm" or "pseudonorm" defined in the way as the note shows in other references? Or are these two concepts equivalent to something more familiar to the people working in functional analysis?
$\def\sp{\kern.4mm}$The semi-pseudonorm in the cited reference is precisely what for example Jarchow (p. 38, Locally Convex Spaces, Teubner 1981) calls an F-seminorm.
Added. (20.9.2015) Besides the one given in Gerald Edgar's answer, another example is $[\sp x\sp]\mapsto\int_0^1|\sp x(t)\sp|\sp(1+|\sp x(t)\sp|\sp)^{-1}{\sp\rm d\sp}t$ in the space $L^0\big([\,0\sp,1\,]\big)$ of all (equivalence classes of real or complex valued) measurable functions on $[\,0\sp,1\,]$ defining the topology for which convergent sequences are precisely those converging in measure.
I guess your example is $\mathcal L^{1/2}[0,1]$: The space of measurable functions $f \;:\; [0,1] \to \mathbb R$ such that the expression $$ p(f) := \int_0^1 |f(t)|^{1/2} dt $$ is finite. And this becomes the space $L^{1/2}[0,1]$ when you identify functions that agree almost everywhere, and you get (4). Another way to treat $L^{1/2}$ is to use $$ q(f) := \big(\int_0^1 |f(t)|^{1/2} dt\big)^2 $$ Then you recover ($1')$ $q(\lambda x) = |\lambda|\;q(x)$ but lose (3) $q(x) + q(y) \le q(x+y)$. Of course both $p$ and $q$ give you the same topololgy for the space.
For the question. I don't know of work using the name "semi-pseudonorm" for this.
For $L^{1/2}$ and friends, maybe try: Kalton, Peck & Roberts, An F-Space Sampler
