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I am studying Linear Algebra Done Right, chapter 2 problem 6 states:

Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.

My solution:

Consider the sequence of functions $x, x^2, x^3, \dots$ This is a linearly independent infinite sequence of functions so clearly this space cannot have a finite basis. However this prove relies on the fact that no $x^n$ is a linear combination of the previous terms. In other words, is it possible for a polynomial of degree $n$ to be equal to a polynomial of degree less than $n$. I believe this is not possible, but does anyone know how to prove this? More specifically, could the following equation ever be true for all $x$?

$x^n = \sum\limits_{k=1}^{n-1} a_kx^k$ where each $a_k \in \mathbb R$

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Then the polynomial $\displaystyle x^n-\sum_{k=0}^{n-1}a_kx^k$ would have infinitely many roots, but it can have $n$, at most.


Another way of dealing with this problem is based upon defining polynomials (in one variable $x$) as expressions of the type $a_0+a_1x+a_2x^2+\cdots+a_nx^n$, where $n\in\{0,1,2,\dots\}$ and each $a_n$ is real. Under this definition, the polynomial $a_0+a_1x+a_2x^2+\cdots+a_nx^n$ is equal to the polynomial $b_0+b_1x+a_2x^2+\cdots+b_nx^n$ if and only if the coefficients are equal, that is, if and only if $a_0=b_0$, $a_1=b_1$, and so on. Under this definition, the problem discussed here is trivial.

What did I prove above then? Well, for each $P(x)\in\mathbb{R}[x]$, there is a corresponding polynomial function from $\mathbb R$ into $\mathbb R$. What I proved above is that this correspondence is one-to-one — when we are dealing with $\mathbb R$. It is still one-to-one if we are dealing with any field with charactristic $0$, such as $\mathbb Q$ or $\mathbb C$. But this is not true in general. For instance, if our field is $\mathbb{F}_2$, then $x$ and $x^2$ are distinct polynomials. But they correspond to the same polynomial function.

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    $\begingroup$ "Equations" can have infinitely many solutions, but nonzero polynomials cannot. $\endgroup$ – Bill Dubuque Aug 6 '17 at 14:11
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    $\begingroup$ @Joe A reference? Why? Isn't my proof clear? $\endgroup$ – José Carlos Santos Aug 6 '17 at 14:14
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    $\begingroup$ @Joe See this answer for a proof of the root bound for polynomials. $\endgroup$ – Bill Dubuque Aug 6 '17 at 14:16
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    $\begingroup$ @Joe You can find several proofs of the Fundamental theorem of algebra. $\endgroup$ – Sahiba Arora Aug 6 '17 at 14:18
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    $\begingroup$ @ZubinMukerjee Jokes aside, I really do mean what I wrote. We should expect statements of results to be precise, but not names of results (else we'd be stuck with monstrous names like "the root bound for nonzero formal univariate polynomials over a commutative integral domain"). $\endgroup$ – Bill Dubuque Aug 6 '17 at 14:59
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By definition, $x^n + \sum\limits_{k=0}^{n-1}a_k x^k$ is a nonzero polynomial of degree $n$. A nonzero polynomial of degree $n$ can have at most $n$ zeros in any field, and $[0,1]$ has more than $n$ elements.

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Two identical functions have identical derivatives. Differentiate your equality $n$ times and you will see the two sides cannot be identical.

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  • $\begingroup$ Good point, but I have not studied derivatives on a rigorous level yet... so I can't really use that as part of my proof. $\endgroup$ – Joe Aug 6 '17 at 15:01
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Suppose $x^n = \sum\limits_{k=1}^{n-1} a_kx^k$. Consider what happens when $x$ is much larger than $|a_1|+\ldots+|a_{n-1}|$. By the triangle inequality, you will get a contradiction. Essentially, you are showing that $x^n$ grows faster than any degree $n-1$ polynomial...

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