I am physicist, learning functional analysis. Why is that separability, completeness etc... Is spoken in terms of Cauchy Sequences for infinite dimensional vector spaces? I cannot get the intuition behind it.
2026-04-06 19:43:14.1775504594
Why are infinite dimensional spaces understood in terms of Cauchy Sequences?
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Euclidean space $\mathbb R^n$ is complete in the sense that every bounded closed set is compact, but for infinite-dimensional spaces being bounded and closed is not equivalent to compactness. This is essentially the reason why one needs to resort to Cauchy sequences.
As an important special case, you could review the construction of the completion of $\mathbb Q$ (namely, $\mathbb R$) in terms of equivalence classes of Cauchy sequences (of rational numbers). This hopefully will give you sufficient feeling for the usefulness of the notion of a Cauchy sequence.