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I noticed that my prof loves to write

$S = span\{v\}$

Instead of $\sum \alpha v$ or $a_1v_1 + a_2v_2+...$. Is he using "span" in a general way? What would span look like in infinite dimensional vector spaces?

DO we represent this using integral or what?

Your neighbor Todorovich
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1 Answers1

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If $V$ is an abstract infinite dimensional vector space, and $\{v_i\}_{i\in I}$ is any collection of vectors, then

$Span\{v_i\}_{i\in I}$ is still the set of all linear combinations of finitely many vectors $\{v_i\}$.

This is because without first specifying some additional structure, vector spaces only guarantee that the sum of finitely many vectors is a vector. It's like in $\mathbb{Z}$, the group operation only says that the addition of two integers is an integer, and hence by induction (and associativity), the addition of finitely many integers is an integer. Thus, the expression $\sum_{n=1}^\infty 1$ is not an integer (it doesn't make sense).

Of course, in $\mathbb{R}$, it's possible to define infinite sums $\sum_{n=1}^\infty a_i$, but only if the series converges. What's going on here is that the addition operation on $\mathbb{R}$ only guarantees that the partial sums $\sum_{n=k}^\infty a_i$ are real numbers. Thus, we have an infinite collection of real numbers:

$$\left\{\sum_{i=1}^k a_i : k\in\mathbb{N}\right\}$$

which are generally speaking all different, and of course generally it wouldn't be right to just define $\sum_{i=1}^\infty a_i$ to be any of the partial sums, so how can we associate a real number to the infinite sum $\sum_{i=1}^\infty a_i$? Well, there are probably a number of ways to do this, but the only way that respects both the topology on $\mathbb{R}$ and the additive group structure on $\mathbb{R}$ is to define

$$\sum_{i=1}^\infty a_i := \lim_{k\rightarrow\infty}\sum_{i=1}^k a_i$$

The key is that on $\mathbb{R}$ you can use its topology to pin down what you mean by convergent infinite sums.

The same can be done with vector spaces. If you specify a topology (ideally a metric topology induced by an absolute value which respects the vector space operations) on a vector space, then you can also define what you mean by a convergent sum of vectors $\sum_{i=1}^\infty v_i$, which will allow you to make sense of certain such infinite sums.

If the vector space is over a topological field like $\mathbb{R}$ or $\mathbb{C}$ then there are a number of topologies you can use - for example, the product topology or the box topology.

EDIT: Nonetheless, as Robert Israel mentioned, the word span always refers to finite linear combinations, whether or not it makes sense to talk about infinite linear combinations.

If you want to read more about this some keywords to google are: normed vector space or Banach space.

oxeimon
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    Just to avoid confusion here: span by itself always refers to linear combinations of finitely many vectors. In a normed vector space, the closure of the span of a set of vectors is called its closed span. But this is not in general the same as the set of convergent infinite sums. – Robert Israel Aug 11 '15 at 22:40
  • Other than infinity, are there examples of "weird" results you might get from adding infinitely many vectors? Is it possible to get something that you wouldn't expect to live in the span? – Thomas Ahle Mar 07 '17 at 06:10
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    @ThomasAhle You could consider the polynomial ring $k[x]$ - it's a $k$-vector space with basis $1,x,x^2,\ldots$. If you let yourself add them all you get $1+x+x^2+\cdots$, which isn't in the vector space - ie, the sum doesn't make sense in the vector space. – oxeimon Mar 09 '17 at 00:21
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    @Nesos Perfect example. Thank you! – Thomas Ahle Mar 09 '17 at 00:23