Clarification of definitions of function spaces with values in another function space I'm reading through Majda and Bertozzi's Vorticity and Incompressible Flow, and in one of the Theorems, they mention some functions being uniformly bounded in the spaces $\text{Lip}([0,T]; H^{m-2}(\mathbb{R}^3))$ and $C_W([0,T]; H^m(\mathbb{R}^3))$. I'd like some clarification on what these two spaces are. 
In the first case, I suppose it's just functions $t \mapsto u(t,x) \in H^{m-2}(\mathbb{R}^3)$ that are Lipschitz continuous. If so, would the norm just be the sup norm from the Hölder Space $C^{0,1}$, i.e., 
$$
\sup_{t\ne s \in [0,T]} \frac{||u(t,x)-u(s,x)||_{H^{m-2}}}{|t-s|},
$$
for $x\in \mathbb{R^3}$ (I suppose this isn't really a norm since $\mathbb{R}^3$ is not bounded)?  
Next, the authors define the space $C_W$ as the continuous functions on $[0,T]$ with values in the weak topology of $H^s$, and further clarify that they mean for any fixed $\varphi\in H^s$, $[\varphi,u(t)]_s$ is a continuous scalar function on $[0,T]$, where 
$$
(u,v)_s = \sum_{|\alpha|\le s} \int_{\mathbb{R}^3} D^\alpha u \cdot D^\alpha v dx.
$$
Here I'm wondering if perhaps there is a typo between the hard and soft brackets. Or since we're looking at $u$ as a function in the weak topology, do the hard brackets represent the dual pairing of $u$ and $\varphi$?
Thank you in advance. The notation and exposition in this book can be a bit confusing and there have been other typos as well, so I'm hoping I can get some clarification.
 A: Your interpretation of $\operatorname{Lip}([0,T]; H^{m-2}(\mathbb{R}^3))$ is correct, except the norm must somehow be inferred from context as there are multiple ways to norm Lipschitz functions. Generally, for a normed space $X$   the space $\operatorname{Lip}([0,T]; X)$ consists of Lipschitz functions $f:[0,T]\to X$. In your case $X$ happens to be  $H^{m-2}(\mathbb{R}^3)$ but the particular space is unimportant. There is a natural semi-norm on $\operatorname{Lip}([0,T]; X)$, namely
$$\sup_{0\le t<s\le T} \frac{\|f(t)-f(s)\|_{X}}{|t-s|}$$ 
but often a norm is desired, in which case one has to add something to give nonzero norm to constants. Popular choices include
$$\|f(0)\|_X + \sup_{0\le t<s\le T} \frac{\|f(t)-f(s)\|_{X}}{|t-s|}$$ 
and
$$\sup_{0\le t\le T}\|f(t)\|_X + \sup_{0\le t<s\le T} \frac{\|f(t)-f(s)\|_{X}}{|t-s|}$$ 

(I suppose this isn't really a norm since $\mathbb{R}^3$ is not bounded)?

I don't see how this is relevant. For any of the above to be a norm, it does not matter what the norm on $X$ is, let alone whether $X$ is a function space on some bounded domain or whatever. We are not taking a supremum over that domain.  
The notation $u(t,x)$ is of course convenient when $f$ takes values in a function space; it replaces the awkward $f(t)(x)$. 

Now we equip $X$ with weak topology and consider $C_W([0,T]; X)$, the space of continuous functions $f:[0,T]\to (X, \mathrm{weak})$. This is precisely the set of functions that that for every fixed $\varphi\in X^*$ (the dual space) the scalar function $t\mapsto \varphi(f(t))$ is continuous. 
Specifically, when $X$ is a Hilbert space of functions, the element $\varphi$ itself is interpreted as an element of $X$ and one writes $[\varphi, f(t)]$ or uses some other brackets which represent the pairing on $X$ that performs an isomorphism of $X$ with its dual space. Yes, there is a typo in the book: the brackets $[\cdot,\cdot]$ and $(\cdot,\cdot)$  should be the same, whichever they use elsewhere.
