# Continuous functional

A continuous operator (continuous mapping) mapping a topological space , which as a rule is also a vector space, into or . Therefore, the definition of, and criteria for, continuity of an arbitrary operator continue to hold for functionals. For example,

1) for a functional , where is a subset of a topological space , to be continuous at a point there must for any be a neighbourhood of such that for (definition of continuity of functionals);

2) a functional that is continuous on a compact set of a separable topological vector space is bounded on this set and attains its least upper and greatest lower bounds (Weierstrass' theorem);

3) since every non-zero linear functional maps a Banach space onto the whole of (or ), it induces an open mapping, that is, the image of any open set is an open set in (or ).

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#### References

[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |

**How to Cite This Entry:**

Continuous functional.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Continuous_functional&oldid=11921