What are the elements of the real vector space $\mathbb{R}[X]$?

Question

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

Answer 1

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 $P(X<x)$ for example, $X$ will always remain $X$, it is the random variable, whereas $x$ can take any real value.