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


In an [answer](https://math.stackexchange.com/a/1095439/9464) to the question in MSE: [The Sobolev Space $H^{1/2}$](https://math.stackexchange.com/q/1095246/9464), $H^{1/2}(\partial\Omega)$ 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): 



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



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> 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. 
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Reference request: definition of $H^{1/2}(\partial\Omega)$ and norm for the image of a bounded linear operator
Note that by this definition the vector space $H^{1/2}(\partial \Omega)$ is isomorphic to the quotient of $H^1(\Omega)$ by the kernel of $\operatorname{tr}$, which you can observe is a closed subspace. In general, given a closed subspace $E$ of a Banach space $X$, the natural "quotient norm" on the quotient $X/E$ is defined by $\|x\|\_{X / E} := \inf\{\|x+y\|\_X : y \in E\}$. This is clearly equivalent to your definition of the $H^{1/2}$ norm.


Quotients of Banach spaces are discussed in most introductory functional analysis textbooks, such as for instance Conway's *Course in Functional Analysis*.


