# Reference request: definition of $H^{1/2}(\partial\Omega)$ and norm for the image of a bounded linear operator

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.

• In your last definition, $Z$ is isomorphic to the quotient of $X$ by the kernel of $\mathcal{L}$, and this is the standard way to put a Banach norm on it. This is discussed in Conway's Course in Functional Analysis, for example. – Nate Eldredge Sep 12 '16 at 2:18
• @NateEldredge: This is exactly what I'm looking for. Thank you very much! – Jack Sep 12 '16 at 2:32

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.
Among the standard definition of fractional Sobolev spaces, $$H^{1/2}(\partial Ω) = \left\{ f \in L^2(\partial Ω) \;|\; \| f \|_{L^2(\partial Ω)} + \int_{\partial Ω}\int_{\partial Ω}\frac{|f(x)-f(y)|^2}{|x-y|^{n+1}} dx \, dy < \infty \right\}$$ and the particular description (Haim Brezis) in terms of Fourier series $$\sum_{\mathbb{Z}}\vert n\vert \vert \hat{f}(n)\vert^2<\infty,$$ one also finds $H^{1/2}$ is defined to be the trace of $H^1$ functions, for example check out Girault, V. and Raviart, P. A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms.