Meaning of $C([0,T], L^2(\Omega))$ In a course of partial differential equations I encounter:
$$f(\mathbf{x},t) \in C([0,T], L^2(\Omega)).$$
Does $C([0,T], L^2(\Omega)) = \{f(\mathbf{x},t)\in L^2(\Omega): \forall t \in [0,T]\}$? 
But what is the meaning of the $C$? I see the notation 
$$f(\mathbf{x},t) \in L^2({]0,T[}, L^2(\Omega))$$
appearing as wel.
 A: Let $V$ be a Banach space, $1 \leq p < \infty$. We define 
$$ L^p(a,b,V)=\{[u]~|~u:[a,b]\to V \text{ is Bochner measurable and satsfies } \int_a^b ||u(t)||^p_{V} dt<\infty \}$$
where Bochner measurable means that there exists a sequence of simple functions $u_n:[a,b]\to V$ such that $u_n(t) \to u(t)$ for $n \to \infty$ for almost all $t \in [a,b]$. This space is a Banach space with the norm $||u||:=\left( \int_a^b ||u(t)||^p_{L^p(V)} dt \right)^{1/p}$.
And to go back to your first question we define
$$C([a,b],V)=\{u ~|~ u:[a,b] \to V \text{ continuous} \}$$
and this is a Banach space with the norm $||u||:=\max_{t \in [a,b]} ||u(t)||_V$. It has some nice properties, for example it can be identified with a closed subspace of the Banach space
$$ L^\infty(a,b,V)=\{[u]~|~u:[a,b]\to V \text{ is Bochner measurable and } \text{ess sup}_{t \in [a,b]} ||u(t)||_V<\infty \text{ holds}\}.$$
For further theory you can have a look at 'Evans: Partial Differential Equations' and 'Wloka: Partial Differential Equations' that have both a great introduction to this topic with several applications to PDEs.
To apply this to your special case we have
$$C([0,T], L^2(\Omega))=\{u ~|~ u:[0,T] \to L^2(\Omega) \text{ continuous} \}$$ so your intuition was quite good but we have additionally continuity.
