An infinite-dimensional subspace of $\mathbb{R}^\infty$ not equal to $\mathbb{R}^\infty$?

Question

Let $V=\mathbb{R}^\infty$ be be the vector space of real-valued sequences. Does there exist an infinite-dimensional subspace $U \subseteq V$ such that $U$ is not equal to $V$?

I'm not even sure where to start with this one and a push in the right direction would be greatly appreciated!

Answer 1

Given a natural number $n$, we can consider the subspace $U$ of $\mathbb{R}^n$ defined by $$U=\{(a_1,\ldots,a_n)\in\mathbb{R}^n:a_1=0\}$$ You should know (and be able to prove) that $\dim(U)=n-1$.

However, if $n$ were not a natural number, but instead $\infty$...

Answer 2

There are several sequence spaces which are quite natural and some of them are quite important in functional analysis.

I will mention just two examples:

• The spaces $c_0$ consisting of all sequences convergent to zero (i.e., all null sequences).
• The space $c_{00}$ consisting of all sequences which have only finitely many non-zero terms. (They are also called sequences with finite support or eventually constant sequences.) You may notice that this is exactly the span of the vectors $e^{i}$, $i\in\mathbb N$, where $e^i$ denotes the sequence which has one on $i$-th coordinate and all other coordinates are zeroes. So, in a sense, this reminds generating $\mathbb R^n$ using standard basis.

It should not be difficult to show that both $c_0$ and $c_{00}$ are infinite-dimensional proper subspaces of what you denote $\mathbb R^\infty$. (For some reasons my personal preference is the notation $\mathbb R^{\mathbb N}$. You can see in some other posts on this site that it is not entirely uncommon notation. This notation has already been mentioned in Peter Franek's comment. Some people also use the notation $\mathbb R^\omega$.)

Basically any sequence space from the the Wikipedia article I linked above should work. (With the caveat that in that article sometimes $\mathbb C$ is used as the base field, so you are interested only in real case.) I have chosen two examples which seemed to be relatively simple and also somewhat natural.

Answer 3

Abstract solution valid for any infinite dimensional space: let be $B$ a basis of $V$. Any infinite proper subset of $B$ spans an infinite dimensional subspace $U$ with $U\ne V$.