What's the definition of the inner product in the trace space $H^{1/2}(\partial\Lambda)$?

Question

Let

• $d\in\mathbb N$
• $\Lambda\subseteq\mathbb R^d$ be bounded and open with Lipschitz boundary

It's well-known that there is a unique bounded linear operator $\gamma_0$ from $H^1(\Lambda,\mathbb R^d)$ to $L^2(\partial\Lambda,\mathbb R^d)$ with $$\gamma_0\left.u\right|_\Lambda=\left.u\right|_{\partial\Lambda}\;\;\;\text{for all }u\in C^1(\overline\Lambda)\;.\tag 1$$ We can show that $$H^{1/2}(\partial\Lambda):=\gamma_0(H^1(D))=\left\{\gamma_0u:u\in H^1(\Lambda)\right\}\tag 2$$ equipped with $$\left\|v\right\|_{H^{1/2}(\partial\Lambda)}:=\inf\left\{\left\|u\right\|_{H^1(\Lambda)}:u\in H^1(\Lambda)\text{ with }\gamma_0u=v\right\}$$ is a $\mathbb R$-Banach space.

Now, I've read that $H^{1/2}(\partial\Lambda)$ is even a $\mathbb R$-Hilbert space. But what's the definition of its inner product?

Remark: I know that the symbol sequence $H^{1/2}(\partial\Lambda)$ already has a well-defined meaning in terms of the fractional order Sobolev spaces. However, since I know nothing about them and $(2)$ is a way to give $H^{1/2}(\partial\Lambda)$ a well-defined meaning too, I hope that we can answer my question without any knowledge abut fractional order Sobolev spaces.

Answer 1

Let $W_0^{1,2}(\Lambda)$ be the kernel of the trace map $\gamma_0 \colon W^{1,2}(\Lambda) \to L^2(\partial \Lambda)$; this coincides with the closure of $C_c^\infty(\Lambda)$, if only $\partial \Lambda$ is smooth enough. The definition $(2)$ you have given is just the quotient space $W^{1,2}(\Lambda) / W_0^{1,2}(\Lambda)$. As you have already noticed, for general Banach spaces the quotient is again a Banach space.

Here $W^{1,2}(\Lambda)$ is a Hilbert space and the quotient can be identified with the orthogonal complement: $$ \left(W_0^{1,2}(\Lambda)\right)^\perp = \bigg\{ u \in W^{1,2}(\Lambda) : u \perp v \text{ for any } v \in W_0^{1,2}(\Lambda) \bigg\}, $$ which of course is a Hilbert space with the inner product inherited from $W^{1,2}(\Lambda)$.

This approach is quite abstract and has nothing to do with the fact that elements of $L^2(\partial \Lambda)$ are functions. It seems to me a bit unproductive unless you are able to define the norm (or the inner product) in an intrinsic way. And this is where fractional Sobolev spaces are needed.

Note. I changed $H^1$ to $W^{1,2}$ to stress that the same can be done for Sobolev spaces $W^{1,p}$ and their trace spaces $W^{1-\frac{1}{p},p}$ (just not in the Hilbert space setting).