PhyslibAlpha.Mathematics.Analysis.Normed.HolderDual

For any real numbers $a$ and $g$ such that $g \ge 0$, the absolute value of the product of the sign of $a$ and $g$ is less than or equal to $g$, written as $|\text{sign}(a) \cdot g| \le g$.

For any natural number $n$, let $p$ and $q$ be real numbers that are Hölder conjugate (satisfying $\frac{1}{p} + \frac{1}{q} = 1$). For any two sequences of real numbers $c = (c_0, \dots, c_{n-1})$ and $v = (v_0, \dots, v_{n-1})$, the absolute value of their dot product is bounded by the product of their $\ell^q$ and $\ell^p$ norms: $$\left| \sum_{i=0}^{n-1} c_i v_i \right| \leq \left( \sum_{i=0}^{n-1} |c_i|^q \right)^{1/q} \left( \sum_{i=0}^{n-1} |v_i|^p \right)^{1/p}$$

For any real numbers $x \ge 0$ and $q > 0$, the inequality $x^{1/q} \le 1$ holds if and only if $x \le 1$.

Let $b = \{b_0, \dots, b_{n-1}\}$ be a basis for a finite-dimensional real vector space $V$. For any vector of coefficients $v = (v_0, \dots, v_{n-1}) \in \mathbb{R}^n$ and any index $i \in \{0, \dots, n-1\}$, the $i$-th coordinate of the linear combination $\sum_{j=0}^{n-1} v_j b_j$ with respect to the basis $b$ is equal to $v_i$.

Let $V$ be a finite-dimensional real normed vector space and $b$ be a basis for $V$. For any continuous linear functional $f: V \to \mathbb{R}$ and any vector $v \in V$, the value of the functional applied to the vector is given by the sum $$f(v) = \sum_{i} f(b_i) \cdot (b.\text{repr}(v))_i$$ where $b_i$ are the basis vectors and $(b.\text{repr}(v))_i$ is the $i$-th coordinate of $v$ in the basis $b$.

Given a basis $b = \{b_i\}_{i \in \{0, \dots, n-1\}}$ of a finite-dimensional real normed vector space $V$ and a sequence of scalars $a = (a_i) \in \mathbb{R}^n$, `HolderDual.ofCoeffs b a` is the unique continuous linear functional $f: V \to \mathbb{R}$ such that $f(b_i) = a_i$ for every $i \in \{0, \dots, n-1\}$.

Let $V$ be a finite-dimensional real normed vector space with a basis $b = \{b_i\}_{i=0}^{n-1}$. For any vector of scalars $a = (a_i)_{i=0}^{n-1} \in \mathbb{R}^n$, the continuous linear functional $f: V \to \mathbb{R}$ defined by the coefficients $a$ relative to the basis $b$ (denoted as `ofCoeffs b a`) satisfies $f(b_i) = a_i$ for every $i \in \{0, \dots, n-1\}$.

Given a basis $b = \{b_i\}_{i \in \{0, \dots, n-1\}}$ of a finite-dimensional real normed vector space $V$, `HolderDual.coeffEquiv b` is the linear equivalence between the continuous dual space $V^* = \mathcal{L}(V, \mathbb{R})$ and the coordinate space $\mathbb{R}^n$. It maps a continuous linear functional $f \in V^*$ to the vector of its values on the basis elements, $(f(b_0), f(b_1), \dots, f(b_{n-1})) \in \mathbb{R}^n$.

Let $V$ be a finite-dimensional real normed vector space with a basis $b = \{b_0, b_1, \dots, b_{n-1}\}$. For any continuous linear functional $f \in V^*$ and any index $i \in \{0, \dots, n-1\}$, the $i$-th component of the coordinate vector associated with $f$ under the linear equivalence `coeffEquiv` is equal to the value of the functional evaluated at the $i$-th basis element, i.e., $(\text{coeffEquiv}_b(f))_i = f(b_i)$.

Let $V$ be a finite-dimensional real normed vector space with a basis $b = \{b_i\}_{i=0}^{n-1}$. Let $\text{coeffEquiv } b: V^* \to \mathbb{R}^n$ be the linear equivalence that maps a continuous linear functional $f \in V^*$ to the vector of its values on the basis elements, $(f(b_0), f(b_1), \dots, f(b_{n-1})) \in \mathbb{R}^n$. For any vector $a \in \mathbb{R}^n$ and any index $i \in \{0, \dots, n-1\}$, the functional $f$ obtained from the inverse mapping, $f = (\text{coeffEquiv } b)^{-1}(a)$, satisfies $f(b_i) = a_i$.

Let $V$ be a finite-dimensional real normed vector space with a basis $b = \{b_i\}_{i=0}^{n-1}$. Let $\Phi : V^* \to \mathbb{R}^n$ be the linear equivalence (denoted as `coeffEquiv b`) that maps a continuous linear functional $f$ to the vector of its values on the basis elements, $(f(b_0), \dots, f(b_{n-1}))$. If there exists a function $N : \mathbb{R}^n \to \mathbb{R}$ such that the norm of any functional $f \in V^*$ is determined by its coordinates as $\|f\| = N(f(b_0), \dots, f(b_{n-1}))$, then the image of the closed unit ball of the dual space $\{f \in V^* \mid \|f\| \leq 1\}$ under $\Phi$ is the set $\{a \in \mathbb{R}^n \mid N(a) \leq 1\}$.

Let $V$ be a finite-dimensional real normed vector space with a basis $\{b_i\}_{i=1}^n$. Suppose there exists a function $N : \mathbb{R}^n \to \mathbb{R}$ such that the norm of any continuous linear functional $f : V \to \mathbb{R}$ is determined by its values on the basis, i.e., $\|f\| = N(f(b_1), \dots, f(b_n))$. Then there is an equivalence between the closed unit ball of the dual space $\{ f \in V \to_L \mathbb{R} \mid \|f\| \leq 1 \}$ and the set of vectors $\{ a \in \mathbb{R}^n \mid N(a) \leq 1 \}$, where each functional $f$ is identified with the vector of its coefficients $a_i = f(b_i)$.

Let $V$ be a finite-dimensional real vector space with a basis $\{b_i\}_{i=1}^n$. Suppose the norm on $V$ is the $\ell^p$ norm with respect to this basis, such that for any vector $v \in V$, $\|v\| = \left( \sum_{i=1}^n |[v]_b^i|^p \right)^{1/p}$, where $[v]_b^i$ denotes the $i$-th coordinate of $v$ in the basis $b$. Then, for any sequence of coefficients $v_j \in \mathbb{R}$, the norm of the linear combination $\sum_{j=1}^n v_j b_j$ is equal to $\left( \sum_{i=1}^n |v_i|^p \right)^{1/p}$.

Let $V$ be a finite-dimensional real vector space with a basis $\{b_i\}_{i=1}^n$. Suppose $V$ is equipped with an $\ell^p$ norm relative to this basis, such that for any $v \in V$, the norm is given by $\|v\| = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p}$, where $x_i$ are the coordinates of $v$ with respect to the basis. Let $p$ and $q$ be Hölder conjugate indices. For any vector $s \in \mathbb{R}^n$ and any sequence of non-negative real numbers $g_i \ge 0$ satisfying $\sum_{i=1}^n g_i^p \le 1$, the norm of the linear combination of basis elements weighted by the sign of $s_i$ and the magnitude $g_i$ is at most $1$: $$\left\| \sum_{i=1}^n (\text{sgn}(s_i) g_i) b_i \right\| \le 1$$

Let $V$ be a finite-dimensional real vector space with a basis $\{b_i\}_{i \in I}$. Let $p$ and $q$ be Hölder conjugate exponents (i.e., $1/p + 1/q = 1$). Suppose $V$ is equipped with the $\ell^p$ norm relative to the basis, such that for any vector $v \in V$, its norm is given by $\|v\| = \left( \sum_i |x_i|^p \right)^{1/p}$, where $x_i$ are the coordinates of $v$ with respect to the basis $\{b_i\}$. For any continuous linear functional $f: V \to \mathbb{R}$, there exists a vector $w \in V$ in the unit ball ($\|w\| \leq 1$) such that $$f(w) = \left( \sum_i |f(b_i)|^q \right)^{1/q}.$$

Let $V$ be a finite-dimensional real vector space with a basis $\{b_i\}$. Suppose the norm on $V$ is defined as the $\ell^p$ norm of the coordinates relative to this basis, such that for any vector $v = \sum_i v_i b_i$, its norm is $\|v\| = \left( \sum_i |v_i|^p \right)^{1/p}$. If $q$ is the Hölder conjugate of $p$ (i.e., $1/p + 1/q = 1$), then for any continuous linear functional $f: V \to \mathbb{R}$, its operator norm $\|f\|$ is equal to the $\ell^q$ norm of its values on the basis: $$\|f\| = \left( \sum_i |f(b_i)|^q \right)^{1/q}$$

Let $V$ be a finite-dimensional real vector space with a basis $\{b_i\}_{i=1}^n$. Suppose $p$ and $q$ are Hölder conjugate exponents (satisfying $1/p + 1/q = 1$). Let $V$ be equipped with the $\ell^p$ norm relative to this basis, such that for any vector $v \in V$, $\|v\| = \left( \sum_{i=1}^n |v_i|^p \right)^{1/p}$ where $v_i$ are the coordinates of $v$ with respect to $\{b_i\}$. Then there is an equivalence between the closed unit ball of the dual space $\{ f \in V \to_L \mathbb{R} \mid \|f\| \leq 1 \}$ and the set of vectors in the closed $\ell^q$ unit ball $\{ a \in \mathbb{R}^n \mid \sum_{i=1}^n |a_i|^q \leq 1 \}$, where a functional $f$ is identified with the vector of its values on the basis $a_i = f(b_i)$.

Let $f: \mathbb{R}^m \to \mathbb{R}$ be a continuous linear functional. For any vector $v \in \mathbb{R}^m$, the value of $f$ at $v$ is given by the sum of the products of its values on the standard basis vectors and the coordinates of $v$: $$f(v) = \sum_{i=1}^m f(e_i) v_i$$ where $e_i$ denotes the $i$-th standard basis vector (the vector with $1$ at index $i$ and $0$ elsewhere) and $v_i$ denotes the $i$-th component of $v$.

For any continuous linear functional $f : \mathbb{R}^m \to \mathbb{R}$ on the space $\mathbb{R}^m$ (equipped with the supremum norm), the operator norm $\|f\|$ is equal to the $\ell^1$ norm of the values of $f$ on the standard basis vectors $e_i$: $$\|f\| = \sum_{i=1}^m |f(e_i)|$$ where $e_i$ denotes the $i$-th standard basis vector (the vector with $1$ at index $i$ and $0$ elsewhere).

Let $f: \mathbb{R}^m \to \mathbb{R}$ be a continuous linear functional, where $\mathbb{R}^m$ is equipped with the supremum norm $\|v\|_\infty = \max_i |v_i|$. Then the operator norm of $f$, denoted $\|f\|$, is equal to the $L^1$ norm of the values of $f$ on the standard basis $e_i$: $$\|f\| = \sum_{i} |f(e_i)|$$ where $e_i$ is the $i$-th standard basis vector (the function that is $1$ at index $i$ and $0$ elsewhere).

Let $\mathbb{R}^m$ be equipped with the supremum norm $\|x\|_\infty = \max_{i} |x_i|$. There is an equivalence between the closed unit ball of the dual space, $\{ f : \mathbb{R}^m \to_L \mathbb{R} \mid \|f\| \leq 1 \}$, and the closed $\ell^1$ unit ball in $\mathbb{R}^m$, $\{ a \in \mathbb{R}^m \mid \sum_{i=1}^m |a_i| \leq 1 \}$. This equivalence identifies a continuous linear functional $f$ with the vector $a$ of its values on the standard basis, $a_i = f(e_i)$.