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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).

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  • $\begingroup$ 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? $\endgroup$
    – LSpice
    Commented Aug 14, 2020 at 14:10
  • $\begingroup$ $\mathcal L^2(\mathbb R) \oplus \mathcal L^2(\mathbb T)$ can be considered as $\mathcal L^2$ of the disjoint union of a line and a circle, with Lebesgue measure on each. $\endgroup$
    – Robert Israel
    Commented Aug 14, 2020 at 14:16
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    $\begingroup$ 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). $\endgroup$
    – Christian Remling
    Commented Aug 14, 2020 at 14:20
  • $\begingroup$ @LSpice Yes, that's right, I will had a precision about that. $\endgroup$
    – Goulifet
    Commented Aug 14, 2020 at 16:48
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    $\begingroup$ @LSpice Sure, I will remember that. $\endgroup$
    – Goulifet
    Commented Aug 21, 2020 at 15:38

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