Category:Continuous Mappings on Normed Vector Spaces

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This category contains results about Continuous Mappings in the context of Normed Vector Spaces.

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 $x \in X_1$.