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    Stone–Weierstrass and a polynomial algebra with high-degree terms

    P-RA-WORKSHOP-D7-W1

    Area
    Real Analysis
    Topics
    Function Spaces, Polynomials, Stone-Weierstrass
    Status
    draft

    Give a precise statement of the Stone–Weierstrass theorem for real-valued continuous functions. Then, verify that the set of all polynomials of the form \[\left\{\sum_{j=2017}^{N}a_jx^j:N\in\mathbb N,\ N\ge2017,\ a_j\in\mathbb R\right\}\] along with the zero function is an algebra over \([-2,2]\subset\mathbb R\).

    Solution

    Goal: (1) State Stone–Weierstrass for real-valued continuous functions; (2) verify that \(\mathcal A = \{\sum_{j=2017}^{N}a_jx^j : N \ge 2017,\ a_j \in \mathbb R\} \cup \{0\}\) is an algebra over \([-2,2]\).

    1. Statement of the theorem (real-valued version).Proof: If \(X\) is a compact Hausdorff space and \(\mathcal A \subseteq C(X;\mathbb R)\) is a subalgebra that (i) contains the constant functions, (ii) separates points of \(X\) (for every \(x \neq y\) there is \(h \in \mathcal A\) with \(h(x) \neq h(y)\)), then \(\mathcal A\) is dense in \(C(X;\mathbb R)\) with the sup norm.

    2. \(\mathcal A\) is the set of all polynomials divisible by \(x^{2017}\) (together with the zero polynomial), restricted to \([-2,2]\).Proof: a polynomial \(p(x) = \sum_{j=0}^{N}a_jx^j\) has no terms of degree \(< 2017\) exactly when \(p(x) = x^{2017}q(x)\) for a polynomial \(q\); the zero polynomial is included explicitly.

    3. \(\mathcal A\) is a vector subspace of \(C[-2,2]\) closed under products.

    3.1. \(0 \in \mathcal A\) by definition.

    3.2. Sums and scalar multiples stay in \(\mathcal A\): \(\sum_{j\ge2017}a_jx^j + \sum_{j\ge2017}b_jx^j = \sum_{j\ge2017}(a_j+b_j)x^j\), and \(c\sum a_jx^j = \sum (ca_j)x^j\).Proof: the lowest degree present remains \(\ge 2017\).

    3.3. Products stay in \(\mathcal A\): \((x^{2017}p)(x^{2017}q) = x^{2017}(x^{2017}pq)\), and the lowest degree of a product of two terms of degree \(\ge 2017\) is \(\ge 2017 + 2017 \ge 2017\).Proof: \(x^{2017}p \cdot x^{2017}q = x^{2017} \cdot [x^{2017}pq]\), and \(x^{2017}pq\) is a polynomial; so the product is of the form \(x^{2017}\cdot(\text{polynomial}) \in \mathcal A\).

    3.4. Q.E.D. Proof: <2>1–<2>3 are the axioms of a subalgebra of \(C[-2,2]\) (scalars over \(\mathbb R\)). No constant functions beyond \(0\) belong to \(\mathcal A\), so the algebra does not contain the constants — it is a subalgebra, and the problem only asks to verify it is an algebra, not that it satisfies the Stone–Weierstrass hypotheses.

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