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.
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 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.