A topological space is locally compact if every point has a neighborhood which is itself contained in a compact set. Many familiar topological spaces are locally compact, including the Euclidean space. Of course, any compact set is locally compact. Some common spaces are not locally compact, such as infinite dimensional Banach spaces. For instance, the L2-space of square integrable functions is not locally compact.

# Locally Compact

## See also

Compact Set, Locally Compact Group, Neighborhood, Topological Space
*This entry contributed by Todd
Rowland*

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## Cite this as:

Rowland, Todd. "Locally Compact." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/LocallyCompact.html