This is cross-posted in MSE (https://math.stackexchange.com/q/1922595/9464) without getting any answer for a while.

In an answer to the question in MSE: The Sobolev Space $H^{1/2}$, $H^{1/2}(\partial\Omega)$ is defined as the range of the trace operator $tr\colon H^1(Ω) \to L^2(\partial Ω)$: $$ H^{1/2}(\partial Ω) = \{ u ∈ L^2(\partial Ω) \;|\; ∃ \tilde u ∈ H^1(Ω)\colon u = tr(\tilde u) \}, \quad \| u \|_{H^{1/2}(\partial Ω)} = \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; tr(\tilde u) = u \}.\tag{*}$$ The domain $\Omega\subset\mathbb{R}^n$ is assumed to be bounded and of class $C^2$. The author of the answer says this might be the most intuitive way to give the definition of $H^{1/2}(\Omega)$. But I'm not able to find a cited reference for the definition above. The only place I can find it is in Temam's Navier Stokes Equations (page 6):

...the space $H^{1/2}(\partial\Omega)$ can be equipped with the norm carried from $H^1(\Omega)$ by $tr$.

It is unclear to me what "equipped with the norm carried from $H^1(\Omega)$ by $tr$" mean.

I'm wondering if this is a very common construction. Could anybody come up with a cited reference in functional analysis regarding the following definition (or the definition (*))?

Let $\mathcal{L}:X\to Y$ be a bounded linear operator between two Banach spaces $X$ and $Y$. Defined $Z:=\mathcal{L}(X)$ and $\|w\|_Z:=\inf\{\|u\|_X\mid \mathcal{L}(u)=w \}$. Then $(Z,\|\cdot\|_Z)$ is a Banach space.

Course in Functional Analysis, for example. $\endgroup$ – Nate Eldredge Sep 12 '16 at 2:18