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We construct a Hilbert space $\\mathcal{W}^{-s}$ for any $s<0$, which contains all the H\\\"older functions of H\\\"older exponents $|s|$ on $ \\mathbb{T}^d\\times \\mathbb{T}^\\ell$. 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We construct a Hilbert space $\\mathcal{W}^{-s}$ for any $s<0$, which contains all the H\\\"older functions of H\\\"older exponents $|s|$ on $ \\mathbb{T}^d\\times \\mathbb{T}^\\ell$. 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