Reference request: norm of the image of a bounded linear operator

Question

In an answer to the question: The Sobolev Space $H^{1/2}$, $H^{1/2}(\partial\Omega)$ is defined as the range of the trace operator $tr\colon H^1(Ω) \to L^2(\partial Ω)$: $$ H^{1/2}(\partial Ω) = \{ u ∈ L^2(\partial Ω) \;|\; ∃ \tilde u ∈ H^1(Ω)\colon u = tr(\tilde u) \}, \quad \| u \|_{H^{1/2}(\partial Ω)} = \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; tr(\tilde u) = u \}.$$ 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 reference for the definition above. I'm wondering if this is a very common construction in functional analysis.

Could anybody come up with a cited reference regarding the following 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.

Answer 1

@Nate gives his answer on MO:

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.