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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!

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    One remark to your notation: it is very common in mathematics to denote by $\Bbb R^\infty$ the space of sequences that only contain finitely many nonzero elements (at least in topology). The space of all real-valued sequences I rather would denote by $\Bbb R^{\Bbb N}$. But it hasn't any impact here. – 2017-02-18
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    The notation was not provided by me, nor is it particularly familiar to me, but thank you for the information :) – 2017-02-18

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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 $\infty$...

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    I'm not sure if I'm quite on the same page as you yet. Do you mean to imply that if V has infinite dimension, we can consider subspace U of infinite size and from there reach the idea that U has infinite dimension? – 2017-02-18
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    I'm suggesting that the subspace $$U=\{(a_1,a_2,\ldots)\in\mathbb{R}^\infty:a_1=0\}$$ of $\mathbb{R}^\infty$ is infinite-dimensional. I don't see what you mean by "size" though - if you mean cardinality (i.e., how many elements are in the set), that's not relevant to the issue of dimension for this problem. – 2017-02-18
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    Okay, thank you. How might I relate this to the problem of whether U=V? – 2017-02-18
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    There are sequences of real numbers that *don't* start with $0$. So the set of sequences of real numbers that *do* start with $0$ can't be all of them... – 2017-02-18
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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 $U\ne V$.

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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.