## Definitions

All of this comes from the same reference[1]

Name Expression Notes
Finite
External direct sum Given $V_1,\cdots,V_n$ which are vector spaces over the same field [ilmath]F[/ilmath]:

$V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)|v_i\in V_i,\ i=1,2,\cdots,n\right\}$
Often written: $V=V_1\boxplus V_2\boxplus\cdots\boxplus V_n$

This is the easiest definition, for example $\mathbb{R}^n=\mathop{\boxplus}^n_{i=1}\mathbb{R}=\underbrace{\mathbb{R}\boxplus\cdots\boxplus\mathbb{R}}_{n\text{ times}}$

Operations: (given [ilmath]u_i,v_i\in V_i[/ilmath] and [ilmath]c[/ilmath] is a scalar in [ilmath]F[/ilmath])

• $(u_1,\cdots,u_n)+(v_1,\cdots,v_n)=(u_1+v_1,\cdots,u_n+v_n)$
• $c(v_1,\cdots,v_n)=(cv_1,\cdots,cv_n)$
Alternative form
$V=\mathop{\boxplus}^n_{i=1}V_i=\left\{\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)\in V_i\ \forall i\in\{1,\cdots,n\}\right\}$ Consider the association:

$(v_1,\cdots,v_n)\mapsto\left[\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)=v_i\ \forall i\right]$
That is, that maps a vector to a function which takes a number from 1 to [ilmath]n[/ilmath] to the [ilmath]i^\text{th} [/ilmath] component, and:
Given a function $f:\{1,\cdots,n\}\rightarrow\cup_{i=1}^nV_i$ where $f(i)\in V_i\ \forall i$ we can define the following association:
$f\mapsto(f(1),\cdots,f(n))$
Thus:

• $V=\mathop{\boxplus}^n_{i=1}V_i=\left\{\left.f:\{1,\cdots,n\}\rightarrow\bigcup_{i=1}^nV_i\right|f(i)\in V_i\ \forall i\right\}$
• $V=\mathop{\boxplus}^n_{i=1}V_i=\left\{(v_1,\cdots,v_n)|v_i\in V_i,\ \forall i\right\}$

Are isomorphic

Sum of vector spaces Given [ilmath]V_1,\cdots,V_n[/ilmath] which are vector subspaces of [ilmath]V[/ilmath]

$\sum^n_{i=1}V_i=\left\{v_1+\cdots+v_n|v_i\in V_i,\ i=1,2,\cdots,n\right\}$
Sometimes this is written: $V_1+V_2+\cdots+V_n$

For any family of vectors (here [ilmath]K[/ilmath] will denote an indexing set and $\mathcal{F}=\left\{V_i|i\in K\right\}$ (a family of vector spaces over [ilmath]F[/ilmath]))
Direct product $V=\prod_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K\right\}$ Generalisation of the external direct sum
External direct sum $V=\mathop{\boxplus}_{i\in K}V_i=\left\{\left.f:K\rightarrow\bigcup_{i\in K}V_i\right|f(i)\in V_i\ \forall i\in K,\ f\text{ has finite support}\right\}$ Note:
• The alternative notation $\bigoplus_{i\in K}^\text{ext}$ is sometimes used
Finite support:
A function [ilmath]f[/ilmath] has finite support if [ilmath]f(i)=0[/ilmath] for all but finitely many [ilmath]i\in K[/ilmath] So it is "zero almost everywhere" - the set $\{f(i)|f(i)\ne 0\}$ is finite.
Internal direct sum Given a family of subspaces of [ilmath](V,F)[/ilmath], $\mathcal{F}=\{V_i|i\in I\}$, the internal direct sum is defined as follows:

$V=\bigoplus\mathcal{F}$ or $V=\bigoplus_{i\in I}$ where the following hold:

1. $V=\sum_{i\in I}V_i$ - that is that [ilmath]V[/ilmath] is the sum (or join) of the family [ilmath]\mathcal{F} [/ilmath]
2. $\forall i\in I$ we have $V_i\cap\left(\sum_{j\ne i}V_j\right)=\{0\}$
• For the second condition each [ilmath]V_j[/ilmath] is called a direct summand of [ilmath]V[/ilmath]
• If [ilmath]\mathcal{F} [/ilmath] is finite, that is $\mathcal{F}=\{V_1,\cdots,V_n\}$ then we often write:
$V=V_1\oplus\cdots\oplus V_n$
• If [ilmath]V=S\oplus T[/ilmath] then we call [ilmath]T[/ilmath] a complement of [ilmath]S[/ilmath] in [ilmath]V[/ilmath]
• The [ilmath]2^\text{nd} [/ilmath] condition is stronger than saying the members of [ilmath]\mathcal{F} [/ilmath] are pairwise disjoint - the book makes this clear although I see it as obvious. (Even though they're not quite pairwise disjoint!)

## References

1. Advanced Linear Algebra - Third Edition - Steven Roman - Graduate Texts in Mathematics