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Complemented subspaces of homogeneous polynomials

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Let \(\mathcal {P}_{K} (^{n}E; F)\) (resp. \(\mathcal {P}_{w} (^{n}E; F)\)) denote the subspace of all \(P\in \mathcal {P}(^{n}E; F)\) which are compact (resp. weakly continuous on bounded sets). We show that if \(\mathcal {P}_{K} (^{n}E; F)\) contains an isomorphic copy of \(c_{0}\), then \(\mathcal {P}_{K} (^{n}E; F)\) is not complemented in \(\mathcal {P}(^{n}E; F)\). Likewise we show that if \(\mathcal {P}_{w} (^{n}E; F)\) contains an isomorphic copy of \(c_{0}\), then \(\mathcal {P}_{w}(^{n}E; F)\) is not complemented in \(\mathcal {P}(^{n}E; F)\).

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Correspondence to Sergio A. Pérez.

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Dedicated to the memory of Jorge Mujica (1946–2017)

Sergio A. Pérez was supported by CAPES and CNPq, Brazil.

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Pérez, S.A. Complemented subspaces of homogeneous polynomials. Rev Mat Complut 31, 153–161 (2018). https://doi.org/10.1007/s13163-017-0240-7

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