Definition:Continuous Mapping (Normed Vector Space)/Space/Definition 1
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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 $X_2$.
$f$ 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 $x \in X_1$.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.2$: Continuous and linear maps. Continuous maps