Difference between revisions of "Pointwise operation"
From Encyclopedia of Mathematics
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(See also: Pointwise order) |
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Extension of an algebraic operation $\star$ on a set $X$ to a set of functions on a set $Y$ taking values in $X$. If $f, g$ are functions taking values in $X$ then the pointwise extension of a [[binary operation]] $\star$ is | Extension of an algebraic operation $\star$ on a set $X$ to a set of functions on a set $Y$ taking values in $X$. If $f, g$ are functions taking values in $X$ then the pointwise extension of a [[binary operation]] $\star$ is | ||
$$ | $$ | ||
f \star g : y \mapsto f(y) \star g(y)\,\ \ \text{for each}\ y \in Y \ . | f \star g : y \mapsto f(y) \star g(y)\,\ \ \text{for each}\ y \in Y \ . | ||
$$ | $$ | ||
| − | The terms "pointwise addition", "pointwise multiplication" are also used. | + | The terms "pointwise addition", "pointwise multiplication" are also used. [[Algebraic operation]]s of different [[signature]] have analogous pointwise extension. |
This may be distinguished from such operations as [[convolution of functions]], where the value of $f*g$ at $y$ does not depend solely on the values $f(y), g(y)$. See also [[Pointwise convergence]]. | This may be distinguished from such operations as [[convolution of functions]], where the value of $f*g$ at $y$ does not depend solely on the values $f(y), g(y)$. See also [[Pointwise convergence]]. | ||
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| + | See also [[Pointwise order]]. | ||
Latest revision as of 19:59, 7 January 2015
2020 Mathematics Subject Classification: Primary: 08-XX [MSN][ZBL]
Extension of an algebraic operation $\star$ on a set $X$ to a set of functions on a set $Y$ taking values in $X$. If $f, g$ are functions taking values in $X$ then the pointwise extension of a binary operation $\star$ is $$ f \star g : y \mapsto f(y) \star g(y)\,\ \ \text{for each}\ y \in Y \ . $$ The terms "pointwise addition", "pointwise multiplication" are also used. Algebraic operations of different signature have analogous pointwise extension.
This may be distinguished from such operations as convolution of functions, where the value of $f*g$ at $y$ does not depend solely on the values $f(y), g(y)$. See also Pointwise convergence.
See also Pointwise order.
How to Cite This Entry:
Pointwise operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_operation&oldid=35264
Pointwise operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pointwise_operation&oldid=35264