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I know that this is a silly question to ask but I would really appreciate if you can answer me. Let $V$ be a vector space and $S=\{v_1,\ldots,v_n\}$ a finite subset of $V$. $S$ is linearly dependent if there exist scalars $a_1,a_2,\ldots,a_n$ not all zero such that $$\sum_{k=1}^n a_kv_k=\vec 0$$

Can we extend this definition if $S$ is infinite? i.e. $S$ is linearly independent if there exist scalars $a_1,a_2,\ldots$ not all zero such that $$\sum_{k=1}^\infty a_k v_k=\vec 0$$

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  • $\begingroup$ I think in the case of $S$ infinite, we still only consider finite sums. $\endgroup$ – angryavian Sep 10 '14 at 1:12
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This is not a silly question. The standard definition is that an infinite set of vectors is linearly dependent if it has some finite subset that is linearly dependent. The definition involves only linear combinations of only finitely many vectors.

But often one deals with "orthonormal bases". An orthonormal basis of an infinite dimensional inner product space is not necessarily a maximal linearly independent set. If it were, then every vector could be written as a linear combination of only finitely many members of the set. In order to speak of infinite linear combinations, one must assign some meaning to the concept of a sequence of vectors in that space converging to a vector in that space. This is defined by letting the distance between $a$ and $b$ be $\|a-b\|=\sqrt{\langle a-b,a-b\rangle}$ where $\langle\cdot,\cdot\rangle$ is the inner product, and then one says the sequence $v_1,v_2,v_3,\ldots$ of vectors converges to $v$ precisely if the sequence $\|v_1-v\|,\|v_2-v\|,\|v_3-v\|,\ldots$ of numbers converges to $0$.

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  • $\begingroup$ Shouldn't it be "if every finite subset of $S$ is linearly dependent" instead of "if there is some finite subset that is linearly dependent"? $\endgroup$ – Vedran Šego Sep 10 '14 at 2:07
  • $\begingroup$ Sorry, typo. It now says "an infinite set $S$ of vectors is linearly dependent if it has some finite subset that is linearly dependent". $\endgroup$ – Michael Hardy Sep 10 '14 at 2:22
  • $\begingroup$ If my standard definition of linear dependence is the one mentioned by OP, how does the fact that 'an arbitrary set $S$ of vectors of a vector space $V$ is linearly dependent in $V$ if there exists a finite subset of $S$ which is linearly dependent on $V$' follow from that definition? Can it be proven? $\endgroup$ – StubbornAtom Aug 26 '16 at 13:57

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