Distribution extension theorem

Question

I'm reading Peter Lax functional analysis. He defined distribution as a continuous element of the dual of the space $ C_0^\infty$ which is the space of all the functions from $R^n$ to C with compact support and infinitely differentiableI. Continuity is defined by sequences and a sequence $u_k$ converge in $C_0^\infty$ if for every multi index a $D^a u_k$ converges uniformly to $D^a u $. I do not understand the 'extension theorem' about linear operator on distribution, that is : Let T be a linear operator from $C_0^\infty $ to $ C_0^\infty$. Then consider the symmetric bilinear form $\int u *v dx$ with u and v element of $C_0^\infty$. He asses that If T' is the traspose of T respect to that bilinear form, for any distribution we can define Tl (a distribution) as (T'v, l) which I do not understand because the integral of T'v *l make no sense for me. Any help is really appreciate

Answer 1

I think that "extension theorem" boils down to the embedding of the space of test functions (or more general of locally integrable functions) into the space of distributions. In the following, I am using L. Schwartz' notation $\mathcal{D}(\mathbb{R}^{n})$ for $C_0^\infty$ and $\mathcal{D}'(\mathbb{R}^{n})$ for the space of distributions on $\mathbb{R}^{n}$.

Given a locally integrable function $f$, for every test function $\varphi\in\mathcal{D}(\mathbb{R}^{n})$ the expression $$ \int_{\mathbb{R}^{n}} \varphi(x)f(x)\;\mathrm{d}x $$ is well-defined. In addition it is not difficult to see that the mapping $$ \mathcal{D}(\mathbb{R}^{n})\to\mathbb{C}, \varphi\mapsto \int_{\mathbb{R}^{n}} \varphi(x)f(x)\,\mathrm{d}x $$ is a continuous linear functional on $\mathcal{D}(\mathbb{R}^{n})$, i.e. it defines an element of $\mathcal{D}'(\mathbb{R}^{n})$ which in the following we denote by $T_f$.

Moreover, given two locally integrable function $f,g$ the distributions defined in the above manner coincide if and only if the mappings $f$ and $g$ are equal almost everywhere. This show that the above construction yields an embedding of the space $L^1_{\mathrm{loc}}(\mathbb{R}^{n})$ of locally integrable functions into the space of distributions.

Note that it is possible to show that under this embedding the space of test functions is dense in the space of distributions. Therefore, if you a mapping can be defined on test functions and is continuous with respect to the topology of $\mathcal{D}'(\mathbb{R}^{n})$, it can be extended to all of $\mathcal{D}'(\mathbb{R}^{n})$.

Using the notation $\langle\cdot,\cdot\rangle$ for the duality mapping between $\mathcal{D}(\mathbb{R}^{n})$ and $\mathcal{D}'(\mathbb{R}^{n})$ for $T\colon\mathcal{D}(\mathbb{R}^{n})\to\mathcal{D}(\mathbb{R}^{n})$ and $\varphi,\psi\in\mathcal{D}(\mathbb{R}^{n})$, we have $$ \langle T\varphi, \psi\rangle = T_\psi(T\varphi) = \int_{\mathbb{R}^{n}} (T\varphi)(x) \psi(x)\;\mathrm{d}x $$ and $$ \langle T\varphi, \psi\rangle = \langle \varphi, T'\psi\rangle = \int_{\mathbb{R}^{n}} \varphi(x) (T'\psi(x))\;\mathrm{d}x $$ by the definition of the transpose.