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# pointwise

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
*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 class 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

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

$\displaystyle(f\cdot g)(x)=$ | $\displaystyle f(x)\cdot g(x)$ | (pointwise multiplication) | |||

$\displaystyle(\lambda f)(x)=$ | $\displaystyle\lambda\cdot f(x)$ | (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 product.

## Mathematics Subject Classification

03-00*no label found*08-00

*no label found*

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