Definition:Continuous Mapping (Normed Vector Space)
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This page is about Continuous Mapping in the context of Normed Vector Space. For other uses, see Continuous Mapping.
Definition
Let $M_1 = \struct{X_1, \norm {\,\cdot\,}_{X_1} }$ and $M_2 = \struct{X_2, \norm {\,\cdot\,}_{X_2} }$ be normed vector spaces.
Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$.
Continuous at a Point
Let $a \in X_1$ be a point in $X_1$.
$f$ is continuous at $a$ (with respect to the norms $\norm {\,\cdot\,}_{X_1}$ and $\norm {\,\cdot\,}_{X_2}$) if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in X_1: \norm {x - a}_{X_1} < \delta \implies \norm {\map f x - \map f a}_{X_2} < \epsilon$
where $\R_{>0}$ denotes the set of all strictly positive real numbers.
Continuous on a Space
$f$ is continuous from $\struct{X_1, \norm {\,\cdot\,}_{X_1} }$ to $\struct{X_2, \norm {\,\cdot\,}_{X_2} }$ if and only if it is continuous at every point $x \in X_1$.
Normed Vector Subspace
Definition:Continuity/Normed Vector Subspace
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.2$: Continuous and linear maps. Continuous maps