definition of linearly dependent set

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 $$\sum_{k=1}^\infty a_k v_k=\vec 0$$

• I think in the case of $S$ infinite, we still only consider finite sums. – angryavian Sep 10 '14 at 1:12

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 $0$.
• 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"? – Vedran Šego Sep 10 '14 at 2:07
• Sorry, typo. It now says "an infinite set $S$ of vectors is linearly dependent if it has some finite subset that is linearly dependent". – Michael Hardy Sep 10 '14 at 2:22
• 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? – StubbornAtom Aug 26 '16 at 13:57