Direct sum/N/Sum/No evaluation/Exercise

Let

${}V=\mathbb {R} ^{(\mathbb {N} )}\,$

be the countably direct sum of ${}\mathbb {R}$ with itself, with the basis ${}e_{n}$, ${}n\in \mathbb {N}$. Let ${}p_{k}$, ${}k\in \mathbb {N}$, be the projections

$\mathbb {R} ^{(\mathbb {N} )}\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto a_{k}.$

a) Show that

$\varphi \colon V\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto \sum _{n\in \mathbb {N} }a_{n},$

is a linear form on ${}V$, which is not a linear combination of the projections.

b) Show that the natural mapping from ${}V$ into its bidual ${}{{V}^{*}}^{*}$ is not surjective.

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