Sum of Two Subspaces and lub - Roman, Chapter 1, page 39

[Math Amateur]

I am reading Steven Roman's book, Advanced Linear Algebra and am currently focussed on Chapter 1: Vector Spaces ... ...

In discussing the sum of a set of subspaces Roman writes (page 39) ...View attachment 5176In the above text, Roman writes:

" ... ... It is not hard to show that the sum of any collection of subspaces of V is a subspace of V and that the sum is the least upper bound under set inclusion ... ... "My questions are as follows:

1) What does Roman mean by the least upper bound of a set of subspaces ... ..

and

2) How do we show that the sum of a collection of subspaces is the lub under set inclusion ...Hope someone can help ...

Peter

 

[Fallen Angel]

Hi Peter,

1) It means that the sum is the minimal subspace that contains all its summands, i. e., it is contained in any other subspace containing that summands.

About 2), a vector $v$ in $S+T$ can be written as $v=s+t$ with $s\in S, t\in T$.
If i have a subspace containing $S$ and $T$, say $W$, since it is closed under addition it must contain any vector of the form $s+t$ with $s\in S$ and $t\in T$.

That's the main idea behind the proof, try to extend it to general sums.

 

[Deveno]

There are certain ideas you are dancing around-perhaps it is time to dive in:

With many kinds of algebraic objects (including: Sets, Groups, Abelian Groups, Rings, Field extensions of a fixed field, $R$-modules, Vector spaces, and Associative Algebras-but this list is not exhaustive) the collection of subobjects forms a LATTICE.

A lattice is a partially-ordered set in which every pair of elements have a (uniquely defined) meet, and join. Meet and join take different "forms" in different lattices, many of which you will recognize:

In the lattice of natural numbers, ordered by divisibility, "meet" is the gcd, and "join" is the lcm.

In the lattice of the power set of a set $S$, $2^S$, ordered by inclusion, meet is intersection, and join is union.

In the lattice of subspaces of a given vector space $V$, meet is intersection, and join is sum.

Some other interesting lattices can be found here: https://en.wikipedia.org/wiki/Lattice_(order)

A generalized meet is often called a greatest lower bound, and a generalized join a least upper bound.