Definition:Central Dilatation Mapping/Definition 1
Definition
Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $\lambda \in K$.
The central dilatation mapping $c_\lambda : X \to X$ is defined as:
- $\forall x \in X: \map {c_\lambda} x = \lambda x$
where $\lambda x$ denotes the scalar product of $\lambda$ with $x$.
Also known as
A central dilatation mapping may also be called the multiplication operator.
It can also be referred to as a dilation mapping, but that can be interpreted ambiguously, as it can also refer to a general dilatation mapping, which includes the possibility of such being a translation.
Some sources use the term enlargement, but that can be misleading as the word is also used when the effect is to shrink the domain.
The terms homothety and similitude can also be seen on occasion.
The adjective homothetic can be used to describe two geometric figures which can be mapped one to the other by a homothety.
Also see
- Results about central dilatation mappings can be found here.
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.6$: Topological vector spaces