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I tried to find a counterexample for it but I failed. If the statement is true, I think it is sufficient to prove that the intersection of two dense subspaces is also dense in the unit ball, but I don't know how to approach it.

fdaeqw
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2 Answers2

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The answer is no. By Stone–Weierstrass theorem a subalgebra of $C([0,1],\mathbb{R})$ is dense if and only if it separates points.

So we need to take two such subalgebras as a counterexample. One will be polynomials. The other one can be for example subalgebra generated by $e^x$, or explicitly all functions of the form $\sum_{k=0}^n a_k e^{kx}$. Their intersection contains constant functions only, which is not dense.

Noah Schweber
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freakish
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In $C[0,1]$ take polynomials of even degrees as a one space, and polynomials of odd degrees as the other. Then each space is linear and dense (the latter follows form the Mutnz theorem for example), yet intersection is empty.

Salcio
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  • Polynomials of odd degree do not form a subspace: $(1+x)+(1-x)=2$. – geetha290krm Oct 22 '23 at 23:13
  • Good point, to correct it let's add constant polynomials to both spaces. – Salcio Oct 23 '23 at 01:42
  • Or still better, let's restrict the space to function which are $0$ at $0$. – Salcio Oct 23 '23 at 01:52
  • @Salcio Doesn't help: $x^3+x^2$ vs. $-x^3+x^2$. Similarly, the polynomials of even degree don't form a subspace via e.g. $x^2+x$ and $-x^2+x$. – Noah Schweber Oct 23 '23 at 01:55
  • These polynomials do not belong to spaces considered ... – Salcio Oct 23 '23 at 03:21
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    Yes, this is correct. The first space is span of monomials $x^{2n}$ and the second one is span of $x^{2n+1}$. In order to avoid constant functions (to meet Müntz–Szász theorem assumptions) we can consider $C[a,b]$ with $a>0$. The only minor thing that needs to be fixed is that their intersection is not empty. It is the trivial zero subspace. – freakish Oct 23 '23 at 07:56
  • @freakish I agree that what you're describing works, but I've only ever seen e.g. "polynomials of even degree" to refer to the set of polynomials whose leading term has even degree (a la wikipedia). – Noah Schweber Oct 24 '23 at 01:46