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For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ a'+a''=a\}. \tag{1} $$

Now, one can show (Lemma 1, [1]) that, for $a\in\ell_2$ this is equal to the dual formulation $$ \lVert a\rVert_{K(t)} = \sup\left\{\sum_{n=1}^\infty a_nb_n : (b_n)_{n\in\mathbb{N}}\in\ell_2,\ \lVert b\rVert_\infty \leq 1,\ \lVert b\rVert_2 \leq t \right\}. \tag{2} $$

My question is: is there a way to relate (2) to (1)? More precisely, is there a correspondence between the $b\in\ell_2$ from (2) and the $(a',a'')\in\ell_1\times\ell_2$ from (1)?


[1] Montgomery-Smith, S. J. (1990). The distribution of Rademacher sums. Proc. Amer. Math. Soc., 109(2), 517–517. doi:10.1090/s0002-9939-1990-1013975-0

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