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