1. **Problem statement:** Given a compact set $K$ in $\mathbb{R}$ disjoint from the interval $I=[0,1]$, and $J$ a minimal closed interval containing both $I$ and $K$. Show for each $n \in \mathbb{N}$ there exists a continuous function $f_n$ such that: - $f_n(x)=0$ for $x \in I$ - $f_n(x)=n+1$ for $x \in K$ - $0 \leq f_n(x) \leq n+1$ for $x \in J$ Further, prove there is a polynomial $P_n$ with real coefficients satisfying: - $|P_n(x)| \leq 1$ for $x \in I$ - $P_n(x) \geq 1$ for $x \in K$ - $|P_n(x)| \leq n+2$ for $x \in J$ Part 1.1.b: Define a norm on space $P$ of polynomials with real coefficients by $$ ||P|| = \sup_{x \in [0,1]} |P(x)| $$ Show this norm defines a normed space structure on $P$. Part 1.1.c: Using part 1.1.a, prove the linear functional $L_a : P \to \mathbb{R}$ given by: $$ L_a(P) = P(a) $$ is continuous if and only if $a \in [0,1]$. Determine necessary and sufficient conditions on $\alpha, \beta$ so that the linear functional: $$ L_{\alpha, \beta}(P) = \int_{\alpha}^{\beta} P(x) dx $$ is continuous. 2. **Problem statement:** Suppose $P_n \subset P$ is the set of all polynomials of degree $\leq n$. Show the natural evaluation map $$ f_p : P_n \to \mathbb{R}^{n+1} $$ given by evaluating polynomial coefficients (or values at $n+1$ points) is an isomorphism from $P_n$ onto $\mathbb{R}^{n+1}$. --- 2. **Step-by-step solution:** ### Part 1.1.a: Constructing continuous functions $f_n$ and polynomials $P_n$ 1. Since $K$ and $I$ are disjoint and compact sets (with $I=[0,1]$), by Urysohn's lemma, there exists continuous functions $f_n: J \to [0,n+1]$ with $f_n|_I = 0$ and $f_n|_K = n+1$. 2. By the Stone-Weierstrass theorem, continuous functions on compact sets can be uniformly approximated by polynomials. Thus, for each $n$, there exists $P_n$ polynomial such that: $$ |P_n(x) - f_n(x)| < 1 \quad \forall x \in J $$ 3. Then, since $f_n(x) = 0$ on $I$, we get $$ |P_n(x)| \leq |P_n(x)-f_n(x)| + |f_n(x)| < 1 + 0 = 1 \quad \forall x \in I $$ 4. For $x \in K$, $f_n(x) = n+1$, so $$ P_n(x) \geq f_n(x) - |P_n(x) - f_n(x)| > (n+1) - 1 = n $$ In particular, $P_n(x) \geq 1$ for $x \in K$. 5. For $x \in J$, $$ |P_n(x)| \leq |P_n(x) - f_n(x)| + |f_n(x)| < 1 + (n+1) = n+2 $$ Thus, all desired inequalities hold. ### Part 1.1.b: Norm on polynomial space $P$ 1. Define norm $$ ||P|| = \sup_{x \in [0,1]} |P(x)| $$ 2. This defines a norm since: - Positivity: $||P|| \geq 0$, and $||P|| = 0$ iff $P=0$ (polynomial zero everywhere means $P=0$). - Homogeneity: For any scalar $c$, $||c P|| = |c| \, ||P||$ by absolute value properties. - Triangle inequality: $||P+Q|| \leq ||P|| + ||Q||$ by supremum and triangle inequality. Hence, $(P,||\cdot||)$ is a normed vector space. ### Part 1.1.c: Continuity of evaluation linear functionals 1. Consider $L_a(P) = P(a)$ for fixed $a$. 2. If $a \in [0,1]$, then $$ |L_a(P)| = |P(a)| \leq ||P|| $$ So $L_a$ is bounded, hence continuous. 3. If $a \notin [0,1]$, choose polynomials that vanish on $[0,1]$ but are arbitrarily large at $a$. Then $L_a$ is not bounded, hence not continuous. 4. For integral linear functional $$ L_{\alpha, \beta}(P) = \int_{\alpha}^{\beta} P(x) dx $$ 5. By definition, $$ |L_{\alpha, \beta}(P)| \leq \int_{\alpha}^{\beta} |P(x)| dx \leq |\beta - \alpha| \, \sup_{x \in [\alpha, \beta]} |P(x)| $$ 6. For $L_{\alpha, \beta}$ to be continuous on $P$ with norm $||P|| = \sup_{x \in [0,1]} |P(x)|$, the interval $[\alpha, \beta]$ must lie within $[0,1]$. Otherwise, $||P|| = \sup_{[0,1]}$ doesn’t control the integral outside. ### Part 2: Isomorphism $f_p$ from $P_n$ to $\mathbb{R}^{n+1}$ 1. $P_n$ is the finite-dimensional vector space of polynomials degree $\leq n$, dimension $n+1$. 2. Define $f_p$ evaluation or coefficient map: $$ f_p: P_n \to \mathbb{R}^{n+1} $$ by mapping polynomial to its coefficients or its values at $n+1$ distinct points. 3. By fundamental theorem of algebra and linear algebra, $f_p$ is a bijective linear map (isomorphism). --- **Final answers:** - Functions $f_n$ and polynomials $P_n$ exist by Urysohn lemma and Stone-Weierstrass approximation, satisfying the bounds. - Supremum norm $||P|| = \sup_{[0,1]} |P(x)|$ defines a norm on polynomials $P$. - Evaluation functionals $L_a$ are continuous iff $a \in [0,1]$. - Integral functionals $L_{\alpha, \beta}$ are continuous iff $[\alpha, \beta] \subseteq [0,1]$. - The evaluation map $f_p: P_n \to \mathbb{R}^{n+1}$ is an isomorphism.