I'm reading Brezzi's paper on DG method and is currently puzzled on how equations (5.7) is used to derive equation (5.8).
Further searches lead me to learning that the space defined $\mathbf{V}'$ is the dual space of $\mathbf{V}$. So let's say that we have $\mathbf{V} = (H^1_0 (\Omega))^2$, then let $\mathbf{v} \in \mathbf{V}$ and $\mathbf{f} \in \mathbf{V}'$. Is there any relationship between their defined inner product:
$$ \left( \mathbf{f}, \mathbf{v} \right) $$
and the norm of $\mathbf{f}$ on $\mathbf{V}'$, i.e. $|| \mathbf{f} ||_{\mathbf{V}'}$? And what space would that inner product be on? The duality pairing of $\mathbf{V} \times \mathbf{V}'$?
I am not having much background in analysis, so any pointers would be much appreciated!
As mentioned in the comments, $(f,v)$ is a duality pairing, where $f\in V'$ and $v\in V$. The key part to get from (5.7) to (5.8) is that the dual norm $$\|f\|_V'=\sup_{w\in V:w\ne 0}\frac{(f,w)}{\|w\|_V},$$ and so we get
$$(f,v) = \frac{(f,v)}{\|v\|_V}\|v\|_{V}\le\left(\sup_{w\in V:w\ne 0}\frac{(f,w)}{\|w\|_V}\right)\|v\|_V=\|f\|_V'\|v\|_V.$$