In wiki I read:
Linear_independence#Infinite_dimensions
But I don't understand what it means $\{x\}_{x \in X} $ and I need examples, and as well as for $(x)_{x \in X} $
Example for $\{x\}_{x \in X} $ and $(x)_{x \in X} $
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$\begingroup$
linear-algebra
abstract-algebra
notation
definition
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3$\{x\}_{x \in X}$ is dumb notation. If interpreted literally, it refers simply to the set $X$. Perhaps $\{x\}_{x \in X}$ is meant to be some kind of "ordered set", or perhaps a set allowing repetition. – 2017-02-12
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0Are you familiar with the idea of a sequence? If so, that will be a useful example here. – 2017-02-12
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0Ok, I am familiar with the idea.. but I don't understand.. – 2017-02-12
1 Answers
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Here's a slightly clearer way to phrase the statement from the page you've linked:
A set $X$ of elements of $V$ is linearly independent if, for some indexing $I$ of the set $X$, the family $\{x_i\}_{i \in I}$ is linearly independent.
Note however, that if we take $I = X$, then we get the statement from the page.
For the purposes of its common usage, there is no distinction between $(x)$ and $\{x\}$. The key is to understand that, in the first definition of linear independence, they assume that the set in question is actually an indexed family, which is to say we have some kind of "ordering" of the elements from our set. Thus, there is the need for a separate proof in the case that a set is not given as an family.
In fact: the distinction is that indices $I$ the family $\{x_i\}_{i \in I}$ are not necessarily ordered, while the indices of the family $(x_i)_{i \in I}$ usually are (that is, $(x)$ is typically used for sequences and nets).