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I am wondering if the Brouwer's fixed point theorem can also be applied in an infinite-dimensional space.

For example let $E = [0, 1] \times [0, 1] \times [0, 1] \times \dots$ be an infinite dimensional space. Let $f$ be a mapping from $E$ into itself such that for any $x \in E$, $$f(x) = (f_1(x), f_2(x), f_3(x), \dots).$$

For any $k = 1, 2, \dots$, I know that $f_k$ is continuous.

Can I say that $E$ is compact and convex and apply the Brouwer's fixed point theorem; ie $f(x) = x$ has a least one solution?

If yes can you give me a reference?

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Brouwer fixed point theorem will not be directly applicable, but some generalisation of it is, in this case the Schauder fixed point theorem. By viewing the set as a compact subspace of the Hausdorff topological vector space $\mathbb{R}^\omega$ (with the product topology), this theorem guarantees the existence of a fixed point.

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