Pointwise2 2005-07-25 11:36:32 2005-07-25 11:36:32 Definition pointwise operation pointwise addition pointwise muliplication % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here When concepts (properties, operations, etc.) on a set $Y$ are extended to functions $f\colon X \longrightarrow Y$ by treating each function value $f(x)$ in isolation, the extended concept is often qualified with the word \emph{pointwise}. One example is pointwise convergence of functions---a sequence $\{f_n\}_{n=1}^\infty$ of functions $X \longrightarrow Y$ converges pointwise to a function $f$ if \(\lim_{n \rightarrow \infty} f_n(x) = f(x)\) for all \(x \in X\). An important \PMlinkescapetext{class} of pointwise concepts are the \emph{pointwise operations}---operations defined on functions by applying the operations to function values separately for each point in the domain of definition. These include \begin{align*} (f+g)(x) ={}& f(x)+g(x) && \text{(pointwise addition)}\\ (f \cdot g)(x) ={}& f(x) \cdot g(x) && \text{(pointwise multiplication)}\\ (\lambda f)(x) ={}& \lambda \cdot f(x) && \text{(pointwise multiplication by scalar)} \end{align*} 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 \emph{not} pointwise is the \PMlinkname{convolution}{Convolution} product.