Definition:Linear Combination of Subsets of Vector Space/Dilation

Definition

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $E$ be a subset of $X$.

Let $\lambda \in K$.

The dilation of $E$ by $\lambda$ is defined and written as:

$\lambda E := \set {\lambda x : x \in E}$

where $\lambda x$ is the scalar product of $x$ by $\lambda$.

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Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra