# The Sobolev Space $H^{1/2}(\partial \Omega)$ as the Quotient Space $H^1(\Omega)/\ker(\text{tr})$

In both questions Reference request: norm of the image of a bounded linear operator and The Sobolev Space $$H^{1/2}$$, the Sobolev space $$H^{1/2}(\partial Ω) = \{ u ∈ L^2(\partial Ω) \;|\; ∃ \tilde u ∈ H^1(Ω)\colon u = \text{tr}(\tilde u) \}$$ is introduced together with the norm $$\| u \|_{H^{1/2}(\partial Ω)} = \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; \text{tr}(\tilde u) = u\}.$$ On the one hand, this space is instantly recognisable as the range of the trace operator, $$\text{tr}$$, but on the other hand, I have read that this space, $$H^{1/2}(\partial\Omega)$$, is constructed as the quotient of $$H^1(\Omega)$$ by $$\ker(\text{tr})$$.

1. The Mathonline page Quotient Normed Linear Spaces defines the quotient normed linear space, for a given normed linear space $$(X,\|\cdot\|_X)$$ and linear subspace $$M\subseteq X$$, as $$X / M := \{ x + M : x \in X \}$$ where $$x + M := \{ x + m : m \in M \}$$. A seminorm is defined on this quotient space by $$\| x + M \|_{X / M} = \inf \{ \| x + m \|_X : m \in M \}$$, which is a genuine norm when $$M\subseteq X$$ is closed. I understand that in our case $$M=\ker(\text{tr})$$, which is closed since $$\text{tr}$$ is a bounded linear operator. Question: why is it that in our case there is no representative $$m$$ of $$M$$ in the definition of $$\| \cdot \|_{H^{1/2}(\partial Ω)}$$?

2. Question: If $$H^{1/2}(\partial Ω)$$ is instantly recognisable as the range of $$\text{tr}$$, how does the need arise to make use of a quotient construction? How is the identification of $$\text{ran}(\text{tr})$$ with the quotient $$H^1(\Omega)/\ker(\text{tr})$$ a natural one to make?

• The answer to part 2. follows (partially) from the first isomorphism theorem: en.wikipedia.org/wiki/Isomorphism_theorems which roughly states that for a group homomorphism (in this case a linear operator between vector spaces) $T : G \to H$ we have that $\mathrm{Im}(T) \cong G/ \ker(T)$. In the case of infinite dimensional spaces the isomorphism is also continuous as well as a vector space homomorphism. Oct 6, 2020 at 20:08

As for point 2, the space was first investigated due to its role as a trace of $$H^1$$-functions, where the quotient construction is a natural abstract characterization of the space. The other characterization was not directly evident, but a result of that investigation, which was a major driving force behind the development of interpolation theory (some historical background can be found in the book "Tartar, L. (2007). An Introduction to Sobolev Spaces and Interpolation Spaces.").