Question

I am told that the term "linear combination" is associated only with finite sets, and I am also told; never to use it with infinite sets.
Can anyone tell me why? I mean why can't I consider the linear combination of a set S, where S is an infinite set.

Answer 1

There is probably a misinterpretation here.

Usually a linear combination refers to a finite number of terms. Whether this finite number of terms is taken from a finite set or infinite set should be irrelevant.

Answer 2

Second that.

Answer 3

Ok, I agree that "a linear combination refers to a finite number of terms". But why?

Answer 4

I mean why can't I consider the linear combination of an infinite number of terms

Answer 5

There's a technical reason for this that isn't obvious in vector spaces like \( \mathbb{R}^n \).

One of the most important collections of vector spaces are vector spaces of functions. An example is the continuous functions over the real numbers, \( C(\mathbb{R}) \).

If we allow an infinite sum of vectors in this space, namely an infinite set of functions, it isn't to hard to construct an infinite sum of continuous functions that is _not_ continuous. (For example, a Fourier series of a step discontinuous function.) In that case, \( C(\mathbb{R}) \) would not be closed under addition and therefore not a vector space.

This is too high a price to pay and for that reason we incorporate into the axioms of the vector spaces that the sum of two vectors must be a vector, an axiom from which you can deduce any finite sum of vetoers must also be a vector; from that it follows that we make no guarantees about infinite some and consequently in other constructions, they are not admitted.

Answer 6

last sentence *infinite SUMS

Answer 7

Wow! You know almost everything. I understand that "C(R) would not be closed under addition and therefore not a vector space. ", and I understand that one of the example is Fourier series of a step discontinuous function

But unfortunately I haven't been taught that yet. So will be waiting till then to taste the example.

Answer 8

I mean I haven't been taught "Fourier series of a step discontinuous function"

Answer 9

and I meant I will get full understanding of this only after I have seen the example and understood. But I feel the reason, thanks!

Answer 10

I have to say I'd never fully thought through the reason infinite sums aren't admitted before you asked the question; but now that I have, I'm quite sure of the answer.

It's a good thing to ask why axioms are what they are, or why a theorem has each of each hypotheses. It makes you a better mathematician.

The unfortunate thing is sometimes you can't see the reason when you first start with something, like this Vector Space example. But still, it was a very good question and I've learnt something as well because you asked it.

Answer 11

Oh really, thanks a lot!

Answer 12

Here's a great quote:

"Don't just read it; fight it! Ask your own questions, look for your own examples,
discover your own proofs. Is the hypothesis necessary? Is the converse true? What
happens in the classical special case? What about the degenerate cases? Where
does the proof use the hypothesis?"

-- Paul Halmos, I Want to be a Mathematician [1985]

Answer 13

Thats wonderful,
I feel that instead of looking at the proof of a theorem directly, if I first try to prove it myself, even though if I turn out to be unsuccessful, I learn a lot about the method used to prove the theorem.

But the worst thing is, I don't get time to do it that way always

Answer 14

Thank you James, I learned something today!