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

Question

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?

Answer 1

I expanded my original answer to the question you linked, hopefully the updated version should clear up the confusion regarding point 1.

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.").