# pointwise

When concepts (properties, operations^{}, etc.) on a set $Y$
are extended to functions^{} $f:X\u27f6Y$
by treating each function value $f(x)$ in isolation, the
extended concept is often qualified with the word
*pointwise*. One example is pointwise convergence
of functions—a sequence^{} ${\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}}$ of
functions $X\u27f6Y$ converges pointwise to
a function $f$ if ${lim}_{n\to \mathrm{\infty}}{f}_{n}(x)=f(x)$
for all $x\in X$.

An important of pointwise concepts
are the *pointwise operations*—operations defined
on functions by applying the operations to function values
separately for each point in the domain of definition. These
include

$(f+g)(x)=$ | $f(x)+g(x)$ | (pointwise addition) | ||

$(f\cdot g)(x)=$ | $f(x)\cdot g(x)$ | (pointwise multiplication^{}) |
||

$(\lambda f)(x)=$ | $\lambda \cdot f(x)$ | (pointwise multiplication by scalar) |

where the identities^{} hold for all $x\in X$. Pointwise
operations inherit such properties as associativity, commutativity,
and distributivity from corresponding operations on $Y$.

An example of an operation on functions which is *not*
pointwise is the convolution (http://planetmath.org/Convolution) product.

Title | pointwise |
---|---|

Canonical name | Pointwise |

Date of creation | 2013-03-22 15:25:00 |

Last modified on | 2013-03-22 15:25:00 |

Owner | lars_h (9802) |

Last modified by | lars_h (9802) |

Numerical id | 4 |

Author | lars_h (9802) |

Entry type | Definition |

Classification | msc 03-00 |

Classification | msc 08-00 |

Defines | pointwise operation |

Defines | pointwise addition |

Defines | pointwise muliplication |