Continuous functional

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

1) for a functional $f : M \rightarrow \mathbf{C}$, where $M$ is a subset of a topological space $X$, to be continuous at a point $x_0 \in M$ there must for any $\epsilon > 0$ be a neighbourhood $U$ of $x_0$ such that $|f(x) - f(x_0)| < \epsilon$ for $x \in U$ (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 $X$ onto the whole of $\mathbf{R}$ (or $\mathbf{C}$), it induces an open mapping, that is, the image of any open set $G \subset X$ is an open set in $\mathbf{R}$ (or $\mathbf{C}$).

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