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Can somebody explain it to me? A few examples of vectors that are elements of $\mathbb{R}[X]$ would be helpful.

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    $R[X]$ is the set of polynomials with real coefficients. So, for example $0.45X^{1000}-X^3- \pi X +2$ is an example of such a vector. – 2017-01-22
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    X is a vector, right? – 2017-01-22
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    Yes, $X$ is a vector, since $X \in R[X]$. – 2017-01-22
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    hm, I see. and when $X^2$ comes, is it the cross product of $X x X$? – 2017-01-22
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    No. In this vecor space there is no cross product, neither scalar product. Not all vector spaces have some cross product: indeed only those of dimension $3$ have, while $R[X]$ has dimension $\infty$. – 2017-01-22
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    and then what is for example $x^3$ in your polynomial? – 2017-01-22
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    Have my explanations brought some light on the subject ? – 2017-01-22

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Uppercase $X$ is called an indeterminate, "invented" to distinguish it from lowercase $x$ which is a variable.

Of course $F(X)=2X^2+3X-7$ is a cousin of $f(x)=2x^2+3x-7$.

But the difference is that $f(x)$ can take values (for example $f(1)=-2$) whereas $f(X)$ will remain... $f(X)$ because $X$ is forbidden to take values.

This may appear as a subtlety. It is not in fact.

Here is a convincing reason, that some abstract algebra texts propose.

Make the association:

$$2X^2+3X-7 \ \ \longleftrightarrow \ \ (-7,3,2,0,0,...)$$

then write:

$$(-7,3,2,0,0,...)=(-7,0,0,0,....)+(0,3,0,0,....)+(0,0,2,0,....)+....$$

Then, introducing the "shift-right operator" $S$:

$$=(-7,0,0,0,0,....)+S[(3,0,0,0....)]+(S \circ S)[(2,0,0,0....)]+...$$

$$=-7(1,0,0,0,0,....)+S[3(1,0,0,0....)]+(S \circ S)[2(1,0,0,0....)]+...$$

$$=(-7+3S+2S^2)[(1,0,0,0,0,....)]$$

Calling this operation $-7+3S+2S^2$ under the name $F(S)$, we are "mimicking" in this way polynomial $F(X)$, but this time with a "dynamical" interpretation.

This situation is "in isomorphy" with the initial polynomial context.

It is this kind of isomorphism that has lead mathematicians to take a more abstract view with indeterminates, etc... at the beginning of XXth century.

Remark 1: In view of the "shift-operator" interpretation given above, $1,X,X^2...$ are sometimes considered as mere "placeholders".

$X^2$, for example, is the third "placeholder" in $(0,0,*,0,...)$ (location of '*').

Remark 2: There is a completely different situation in which there is a distinction between lowercase and uppercase letters, it is random variables in pobability: when I writes $P(X

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    thanks JeanMarie, it helped :) – 2017-01-22
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    for example if $U:{(f∈R[X]∣deg(f)=3)}$ then U is the aggregation of all 3-dimensional Vectors? So I could also write $U=R^3$? – 2017-01-22
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    First of all: it is $U:{(f∈R[X]∣deg(f) \leq 3)}$ which is a vector space. $U:{(f∈R[X]∣deg(f)=3)}$ is not a vector space because for example the sum of $X^3$ and $-X^3$ is not in $U$. 2nd remark : $U$ is 4 dimensional with basis $\{1,X,X^2,X^3\}$ – 2017-01-22