David Hilbert's Problem #5

Hilbert’s Fifth Problem

Understanding Lie Groups: Are continuous groups automatically differential groups

Hilbert’s fifth problem concerns Lie groups, which are algebraic objects that describe continuous transformations. Hilbert’s question is whether Lie’s original framework, which assumes that certain functions are differentiable, works without the assumption of differentiability. In 1952, Andrew Gleason, Deane Montgomery and Leo Zippin answered the question, showing that the same theory arises whether differentiability is assumed or not. Some mathematicians have interpreted the question differently and consequently have different answers.

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