Definition:Linear Combination of Subsets of Vector Space/Binary Case

Definition

Let $K$ be a field.

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

Let $A$ and $B$ be subsets of $X$.

Let $\lambda, \mu \in K$.

We define the linear combination $\lambda A + \mu B$ by:

$\lambda A + \mu B = \set {\lambda a + \mu b : a \in A, \, b \in B}$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra
• 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $2.1$: Normed Spaces