Brouwer's fixed point theorem in an infinite-dimensional space

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?

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.