Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value 
of some function 
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. Important relations can also be defined pointwise.
Pointwise operations
Examples include
where 
.
See pointwise product, scalar.
Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. An example of an operation on functions which is not pointwise is convolution.
By taking some algebraic structure 
in the place of 
, we can turn the set of all functions 
to the carrier set of into an algebraic structure of the same type in an analogous way.
Componentwise operations
Componentwise operations are usually defined on vectors, where vectors are elements of the set 
for some natural number 
and some field 
. If we denote the 
-th component of any vector 
as 
, then componentwise addition is 
.
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