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

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.

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.