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I was told that every vector space can be written as a directsum of fields. However I don't see how this could be true for the space of all functions $f:\mathbb{R}\to\mathbb{R}$ (with addition and multiplication defined pointwise). Could someone explain to me how this space could be represented as a direct sum?

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Let $V$ be any vector space over a field $F$, and let $B$ be a (Hamel) basis. Then by definition each $v\in V$ can be expressed uniquely in the form $\sum_{b\in B}\alpha_bb$, where each $\alpha_b\in F$, and all but finitely many of the coefficients $\alpha_b$ are $0_F$. For each $b\in B$ let $F_b$ be a copy of $F$; then

$$\varphi:\bigoplus_{b\in B}F_b\to V:\langle \alpha_b:b\in B\rangle\mapsto\sum_{b\in B}\alpha_bb$$

is an isomorphism. In the infinite-dimensional case the existence of a Hamel base follows from the axiom of choice, and in general you won’t be able to write one down explicitly.

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