# Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $$L^2(\mathbb{R})$$ (square-integrable functions) and $$L^2(\mathbb{T})$$ (1-periodic square-integrable functions, considered over the real line $$\mathbb{R}$$) are two subspaces of the space of tempered distributions $$\mathcal{S}'(\mathbb{R})$$ and one can easily show that the sum $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$ is direct.

The duals of $$L^2(\mathbb{R})$$ and $$L^2(\mathbb{T})$$ are isometrically isomorphic to $$L^2(\mathbb{R})$$ and $$L^2(\mathbb{T})$$, respectively (Riesz representation theorem). Therefore, the continuous dual of the direct sum is simply $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$ in the sense that (1) an element $$g_1 + g_2 \in L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$ defines a continuous linear functional over $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$ via $$(f_1 + f_2) \mapsto \langle f_1 , g_1 \rangle_{L^2(\mathbb{R})} + \langle f_2 , g_2 \rangle_{L^2(\mathbb{T})}$$ (which uses that both decompositions $$f = f_1 + f_2$$ and $$g = g_1+g_2$$ are unique), and that (2) any element of $$(L^2(\mathbb{R}) \oplus L^2(\mathbb{T}))')$$ is of this form.

I would like to identify the subset $$\mathcal{X}\subset \mathcal{S}'(\mathbb{R})$$ of functions $$g$$ such that $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T}) \ni f_1 + f_2 \mapsto \int_{\mathbb{R}} g(x) (f_1 + f_2)(x)\mathrm{d}x$$ specifies a continuous linear functional over $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$. Clearly, by restricting it to $$L^2(\mathbb{R})$$ (i.e. setting $$f_2=0$$), we need to have $$g \in L^2(\mathbb{R})$$. Moreover, $$\mathcal{X}$$ contains any square-integrable compactly supported functions, but also functions that are not compactly supported but that have sufficiently nice asymptotic properties such that the integral $$\int_{\mathbb{R}} g (x) f_2(x)\mathrm{d}x$$ is well-defined for any square-integrable periodic $$f_2$$ and defines a continuous functional over $$L^2(\mathbb{T})$$.

Question: Is there a way to identify the space $$\mathcal{X}$$ I am depicting? Can we reach any linear functionals over $$L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$$ by doing so? I am also interested by generalization to other direct sums between spaces of periodic and non-periodic functions (e.g., $$L^p$$-spaces, or spaces of continuous-functions for the supremum norm).

• To regard $\mathcal L^2(\mathbb T)$ as a subspace of the tempered distributions, you pull back to periodic functions on $\mathbb R$ and then integrate them against Schwartz functions? – LSpice Aug 14 at 14:10
• $\mathcal L^2(\mathbb R) \oplus \mathcal L^2(\mathbb T)$ can be considered as $\mathcal L^2$ of the disjoint union of a line and a circle, with Lebesgue measure on each. – Robert Israel Aug 14 at 14:16
• What do you mean by $f_1+f_2$ in the last displayed equation? There doesn't seem to be an obvious way to define this sum that gives an element of $L^2(\mathbb R)$ (which is what you need here). – Christian Remling Aug 14 at 14:20
• @LSpice Yes, that's right, I will had a precision about that. – Goulifet Aug 14 at 16:48
• @LSpice Sure, I will remember that. – Goulifet Aug 21 at 15:38