Finite and Infinite Bases

please tell me how $$\mathbb{R}^n$$ has finite basis ? I understand the polynomial part but not the $$\mathbb{R}^n$$.

• It means a basis has finitely many elements. Commented Nov 18, 2014 at 19:18
• i do understand that , but how Rn has finite basis ?? Commented Nov 18, 2014 at 19:18
• @MohamedOsama: Do you agree that $\mathbb R^2$ has a basis with two elements, for example? Such as $\{(1,0),(0,1)\}$. Commented Nov 18, 2014 at 19:20
• Ever consider trying to find a basis for Rn that has more than n vectors? Give it a try. Commented Nov 18, 2014 at 19:20
• yes i do it has dimension = 2 Commented Nov 18, 2014 at 19:21

What it says is that for every $n\ge 0$ the space $\mathbb R^n$ has a finite basis.
There is not a single space called "$\mathbb R^n$" in and of itself -- there is only $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth. Speaking about $\mathbb R^n$ is just a way to make the same claim about each of these spaces separately.
• @MohamedOsama: There is one space of all polynomials. That is a completely different situation from $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth being different vector spaces. Commented Nov 18, 2014 at 19:27
• @MohamedOsama: He's using standard terminology. Saying "$\mathbb R^n$ is finite dimensional" is neither more nor less than saying "Each of $\mathbb R$, $\mathbb R^2$, $\mathbb R^3$ and so forth is finite dimensional". Commented Nov 18, 2014 at 19:32
• On the other hand, saying something about $P$ is saying something about one particular vector space (and only that), namely the one that consists of all polynomials. Commented Nov 18, 2014 at 19:34