how do we prove that an infinite set of functions given for example as: $g \cup f_i, \text{ where } g(n)=1, f_i(i)=1 \text{ and } f_i(n)=0 \text{ for all } i \neq n, \forall i,n \in \mathbb {N}$ is lineary independent.

I can see that vectors from f have a form of (0,0,0,0,1....0,0,0) with 1 in the index where i=n. But I don´t quite understand the relation to g. Following this logic, it would seem that vectors from g are only 1-s. Are those 1-s from g not a linear combination of the vectors from f? I can´t quite understand why it is linearly independent!

  • $\begingroup$ Linear dependency means that you have a finite sum of $f_i$, which is obviously not possible. $\endgroup$ – sehigle Apr 28 at 17:00
  • $\begingroup$ What about g? How can I visualize g? f could be visualized as the set of canonic vectors. But what about g? $\endgroup$ – infinitedreamer666 Apr 28 at 18:05
  • $\begingroup$ I would characterize $g$ as the limit of $\sum_{i<n} f_i$ for $n\rightarrow\infty$. $\endgroup$ – sehigle Apr 28 at 18:31

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