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Suppose $\left\{ V_{i}\right\}_{i\in I}$ is a collection of vector spaces over $K$. Define $$\prod _{i\in I}V_{i}=\left\{ f:I\rightarrow \bigcup _{i\in I}V_{i}|f\left( i\right) \in V_i\right\}$$ where $$\left( f+g\right) \left( i\right) =f\left( i\right) +g\left( i\right)$$

$$\left( \lambda f\right) \left( i\right) =\lambda \left( f\left( i\right) \right).$$

Then $\prod _{i\in I}V_{i}$ is a vector space over $K$.

What is the notation of $\bigcup _{i\in I}V_{i}$ mean? and can you give an example and can you explain this?

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    Maybe you could be a little more explicit about what your question is? – 2017-02-26
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    This vectorspace is the direct product of vectorspaces. In the finite case (and even in the countable infinite case), you would denote the elements of this vectorspace as $n$-tuples (or in the countable infinite case: sequences). However, in the uncountable case, you have to work around this and therefore this definition of 'maps' is used. It maps every index in $I$ to a 'coordinate' $f(i)$ which is an element of the corresponding vectorspace $V_i$. If this confuses you, you could just consider them being tuples, although this is (strictly) not correct. – 2017-02-26
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    @ziggurism edited. – 2017-02-26
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    $\cup V_i$ is the union of the sets $V_i$. Union means take elements from any of the summands. For example, the union $\{1,2\}\cup\{7,8\}=\{1,2,7,8\}$. And if you want to take the union of vector spaces, it is the same. – 2017-02-26
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    @ziggurism can you give an example union of vector spaces? – 2017-02-26
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    There's not much more to say. For example if $V_1=\langle e_1,e_2\rangle$ and $V_2=\langle u_1,u_2\rangle$, then the union is vectors from either vector space. Many vector spaces are infinite, so I don't list all the elements, just the generators. Note that the union of two vector spaces is not itself a vector space; it does not include sums of vectors from the different spaces. – 2017-02-26

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