Proof of Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones

Each result cited below is universally quantified over the data in its own statement.

Conventions. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, $E_{d}$ with pointwise operations is a real inner product space with norm $\lVert\cdot\rVert_{d}$; in particular it is a vector space over $\mathbb{R}$, and its additive inverse of $y$ is $(-1)y$ by claim 5 of Elementary Identities in a Vector Space, so the difference $x-y=x+(-1)y$ of The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge §space is the vector-space difference $x+(-y)$. The identities of Elementary Identities in a Real Inner Product Space (bilinearity, $|\lambda x|=|\lambda|\,|x|$ and $\lVert-x\rVert_{d}=\lVert x\rVert_{d}$ from its clause homogeneity), the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and the Cauchy–Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space are used for $E_{d}$ without further comment on their applicability. Order manipulations in $\mathbb{R}$ are those of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field; absolute values are handled by Properties of the Absolute Value in an Ordered Field. Write $x_{0}=\iota_{d}(\lambda)$.

Step 1 (the core estimate). We show: if $G:E_{d}\to\mathbb{R}$ and $q\in E_{d}$ are such that $G$ is differentiable at $x_{0}$ with gradient $q$ in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable (with $E=U=E_{d}$), and $G\circ\iota_{d}$ has a local minimum at $\lambda$ relative to $\mathcal{L}_{d}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$ (Local Minimum of a Function Relative to a Subset of a Metric Space), then $0\le\langle q,v\rangle_{d}$ for every $v\in T^{+}_{\lambda}$.

By Local Minimum of a Function Relative to a Subset of a Metric Space, fix a real $\delta_{0}>0$ such that $G(\iota_{d}(\lambda))\le G(\iota_{d}(\mu))$ for every $\mu\in\mathcal{L}_{d}$ with $d_{\mathcal{L}}(\lambda,\mu)<\delta_{0}$. Let $v\in T^{+}_{\lambda}$ and, by Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws §inward, fix $s_{0}>0$ and $\gamma$ as there. For real $s$ with $0<s\le s_{0}$ put

$$z_{s}=\iota_{d}(\gamma(s))-x_{0}\in E_{d},\qquad e_{s}=\tfrac{1}{s}z_{s}-v\in E_{d}.$$

By the vector-space axioms, $s(v+e_{s})=sv+s\cdot\tfrac1s z_{s}-sv=z_{s}$ and $x_{0}+z_{s}=\iota_{d}(\gamma(s))$.

Put $C=\lVert v\rVert_{d}+1+\lVert q\rVert_{d}$ and $K=\lVert v\rVert_{d}+1$; both are positive, since norms are nonnegative and $0<1$ (claim 6 of Elementary Order Arithmetic in an Ordered Field). Let $t>0$ be real. Choices are made in the following order. Let $\varepsilon'$ be the lesser of $t\,C^{-1}$ and $1$ (claim 9 of Elementary Order Arithmetic in an Ordered Field); it is positive, being one of these two positive numbers (claims 7 and 5 of that lemma). By Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws §inward, applied with $\varepsilon'$, choose a real $\delta_{T}>0$ with $\lVert e_{s}\rVert_{d}\le\varepsilon'$ for every real $s$ with $0<s\le s_{0}$ and $s<\delta_{T}$. By Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable, applied with $\varepsilon'$, choose a real $\delta_{F}>0$ such that every $z\in E_{d}$ with $\lVert z\rVert_{d}<\delta_{F}$ satisfies $|G(x_{0}+z)-G(x_{0})-\langle q,z\rangle_{d}|\le\varepsilon'\lVert z\rVert_{d}$. Let $m$ be the least of $s_{0}$, $\delta_{T}$, $\delta_{F}K^{-1}$ and $\delta_{0}K^{-1}$ (repeated use of claim 9 of Elementary Order Arithmetic in an Ordered Field), which is positive, being one of these positive numbers, and put $s=m\cdot2^{-1}$. By claim 8 of that lemma, $0<s<m$, so $0<s\le s_{0}$, $s<\delta_{T}$, $sK<\delta_{F}$ and $sK<\delta_{0}$ (the last two by claim 10 of that lemma, multiplying $s<\delta_{F}K^{-1}$ and $s<\delta_{0}K^{-1}$ by $K>0$).

For this $s$: $\lVert e_{s}\rVert_{d}\le\varepsilon'\le1$, so $\lVert v+e_{s}\rVert_{d}\le\lVert v\rVert_{d}+\lVert e_{s}\rVert_{d}\le K$, and $\lVert z_{s}\rVert_{d}=\lVert s(v+e_{s})\rVert_{d}=s\lVert v+e_{s}\rVert_{d}\le sK$, as $|s|=s$. Hence $\lVert z_{s}\rVert_{d}<\delta_{F}$ and $\lVert z_{s}\rVert_{d}<\delta_{0}$. By The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance, $d_{\mathcal{L}}(\lambda,\gamma(s))=\lVert x_{0}-\iota_{d}(\gamma(s))\rVert_{d}=\lVert-z_{s}\rVert_{d}=\lVert z_{s}\rVert_{d}<\delta_{0}$, since $x_{0}-\iota_{d}(\gamma(s))=-z_{s}$ in the vector space $E_{d}$. The choice of $\delta_{0}$ gives $G(x_{0})\le G(\iota_{d}(\gamma(s)))=G(x_{0}+z_{s})$, and the choice of $\delta_{F}$ gives $G(x_{0}+z_{s})-G(x_{0})-\langle q,z_{s}\rangle_{d}\le\varepsilon'\lVert z_{s}\rVert_{d}$ (claim 3 of Properties of the Absolute Value in an Ordered Field). Adding $\langle q,z_{s}\rangle_{d}-\varepsilon'\lVert z_{s}\rVert_{d}$ to both sides of the second inequality (claim 3 of Elementary Arithmetic in an Ordered Field in its translated form) gives $\langle q,z_{s}\rangle_{d}\ge\bigl(G(x_{0}+z_{s})-G(x_{0})\bigr)-\varepsilon'\lVert z_{s}\rVert_{d}$, and the first inequality gives $0\le G(x_{0}+z_{s})-G(x_{0})$; hence

\langle q,z_{s}\rangle_{d}\ge-\varepsilon'\lVert z_{s}\rVert_{d}. $$ By [Elementary Identities in a Real Inner Product Space §bilinear](/theorems/5ce8e666-53b7-4958-beed-be505dcf38ba?v=5bbf7092-58eb-4039-aad3-ddfbf22d02dd#clause-bilinear), $\langle q,z_{s}\rangle_{d}=\langle q,s(v+e_{s})\rangle_{d}=s\bigl(\langle q,v\rangle_{d}+\langle q,e_{s}\rangle_{d}\bigr)$, and $\lVert z_{s}\rVert_{d}=s\lVert v+e_{s}\rVert_{d}$. Multiplying the inequality $s\bigl(\langle q,v\rangle_{d}+\langle q,e_{s}\rangle_{d}\bigr)\ge-\varepsilon' s\lVert v+e_{s}\rVert_{d}$ by $s^{-1}>0$ (claim 5 of [Elementary Arithmetic in an Ordered Field](/theorems/fe1c552a-759a-469b-86a4-5207560aefaf?v=63c2af8b-b56f-40d4-bc16-3fb0690235bc)),

\langle q,v\rangle_{d}\ge-\varepsilon'\lVert v+e_{s}\rVert_{d}-\langle q,e_{s}\rangle_{d}\ge-\varepsilon'K-\lVert q\rVert_{d},\varepsilon'=-\varepsilon'C\ge-t,

using $\varepsilon'\lVert v+e_{s}\rVert_{d}\le\varepsilon'K$ (multiply $\lVert v+e_{s}\rVert_{d}\le K$ by $\varepsilon'\ge0$, claim 5 of [Elementary Arithmetic in an Ordered Field](/theorems/fe1c552a-759a-469b-86a4-5207560aefaf?v=63c2af8b-b56f-40d4-bc16-3fb0690235bc)), $\langle q,e_{s}\rangle_{d}\le|\langle q,e_{s}\rangle_{d}|\le\lVert q\rVert_{d}\lVert e_{s}\rVert_{d}\le\lVert q\rVert_{d}\varepsilon'$ ([The Cauchy-Schwarz Inequality in a Real Inner Product Space](/theorems/75e68dc9-c425-4d4e-afd8-53eba404a50e?v=638ad390-44c9-472a-bccb-9ba6242bb219), and claim 5 of [Elementary Arithmetic in an Ordered Field](/theorems/fe1c552a-759a-469b-86a4-5207560aefaf?v=63c2af8b-b56f-40d4-bc16-3fb0690235bc) with $\lVert q\rVert_{d}\ge0$), and $\varepsilon'C\le tC^{-1}C=t$ (multiply $\varepsilon'\le tC^{-1}$ by $C\ge0$, claim 5 of [Elementary Arithmetic in an Ordered Field](/theorems/fe1c552a-759a-469b-86a4-5207560aefaf?v=63c2af8b-b56f-40d4-bc16-3fb0690235bc)). Since $t>0$ was arbitrary, $0-t\le\langle q,v\rangle_{d}$ for every positive $t$, and [Comparison of Real Numbers with Arbitrary Positive Slack §slack-below](/theorems/9f05afaa-f936-456f-82da-3660b1ae576a?v=94e2e22c-980b-43b9-8911-f6a0a2914c70#clause-slack-below) gives $0\le\langle q,v\rangle_{d}$. This proves Step 1. **Step 2 (two-sided vectors).** Let $v\in T^{\pm}_{\lambda}$, and fix $s_{0}>0$ and $\gamma$ as in [Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws §two-sided](/theorems/508a4beb-d46f-44fe-b806-0fb4f69cd059?v=11018f9f-faa5-4c63-ad15-5a4f3046be06#clause-two-sided). (a) $v\in T^{+}_{\lambda}$: take the restriction of $\gamma$ to $\{s:0\le s\le s_{0}\}$; for $\varepsilon>0$, the $\delta$ of the two-sided condition works, since $0<s\le s_{0}$ and $s<\delta$ give $0<|s|\le s_{0}$ and $|s|<\delta$. (b) $-v=(-1)v\in T^{+}_{\lambda}$: let $\gamma^{-}(s)=\gamma(-s)$ for $0\le s\le s_{0}$, which is defined since $-s_{0}\le-s\le0$, with $\gamma^{-}(0)=\lambda$. For $\varepsilon>0$ let $\delta$ be as in the two-sided condition, and let $0<s\le s_{0}$ with $s<\delta$; with $\sigma=-s$ one has $0<|\sigma|=s\le s_{0}$ and $|\sigma|<\delta$, and $\tfrac1s=-\tfrac1\sigma$, so

\Bigl\lVert\tfrac{1}{s}\bigl(\iota_{d}(\gamma^{-}(s))-x_{0}\bigr)-(-v)\Bigr\rVert_{d}=\Bigl\lVert-\Bigl(\tfrac{1}{\sigma}\bigl(\iota_{d}(\gamma(\sigma))-x_{0}\bigr)-v\Bigr)\Bigr\rVert_{d}=\Bigl\lVert\tfrac{1}{\sigma}\bigl(\iota_{d}(\gamma(\sigma))-x_{0}\bigr)-v\Bigr\rVert_{d}\le\varepsilon .

Hence, under the hypotheses of Step 1, $0\le\langle q,v\rangle_{d}$ and $0\le\langle q,-v\rangle_{d}=-\langle q,v\rangle_{d}$ ([Elementary Identities in a Real Inner Product Space §bilinear](/theorems/5ce8e666-53b7-4958-beed-be505dcf38ba?v=5bbf7092-58eb-4039-aad3-ddfbf22d02dd#clause-bilinear)), so $\langle q,v\rangle_{d}\le0$ by claim 4 of [Elementary Order Arithmetic in an Ordered Field](/theorems/76f88645-0184-4f0e-a3c9-f9967fffeea7?v=64306ed0-32ee-43f6-a710-e12e0d00ce51), and $\langle q,v\rangle_{d}=0$ by antisymmetry of the order $\le$ of $\mathbb{R}$, which is a [total order](/theorems/58872eb3-b645-437b-90d4-344ced6a8842?v=8139d6a0-e5f1-4424-b52e-05a3fa204a1b). **Clause 1.** By [Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient](/theorems/27523721-62a4-4363-8a50-9327cbb20d36?v=da1af566-e7ea-4ae9-b8cb-ddaa6ac67af9#clause-gradient), $F$ is differentiable at $x_{0}$ with gradient $DF(x_{0})$ in the sense of [Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable](/theorems/27523721-62a4-4363-8a50-9327cbb20d36?v=da1af566-e7ea-4ae9-b8cb-ddaa6ac67af9#clause-differentiable). If $F\circ\iota_{d}$ has a local minimum at $\lambda$ relative to $\mathcal{L}_{d}$, Steps 1 and 2 with $G=F$ and $q=DF(x_{0})$ give $0\le\langle DF(x_{0}),v\rangle_{d}$ for $v\in T^{+}_{\lambda}$ and $\langle DF(x_{0}),v\rangle_{d}=0$ for $v\in T^{\pm}_{\lambda}$. **Clause 2.** Suppose $F\circ\iota_{d}$ has a local maximum at $\lambda$ relative to $\mathcal{L}_{d}$, and put $G(x)=-F(x)$ for $x\in E_{d}$ and $q=-DF(x_{0})$. (i) $G$ is differentiable at $x_{0}$ with gradient $q$: given $\varepsilon>0$, let $\delta>0$ be as in [Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable](/theorems/27523721-62a4-4363-8a50-9327cbb20d36?v=da1af566-e7ea-4ae9-b8cb-ddaa6ac67af9#clause-differentiable) for $F$ and $DF(x_{0})$; for $z\in E_{d}$ with $\lVert z\rVert_{d}<\delta$ one has $x_{0}+z\in E_{d}$ and, since $\langle-DF(x_{0}),z\rangle_{d}=-\langle DF(x_{0}),z\rangle_{d}$ by [Elementary Identities in a Real Inner Product Space §bilinear](/theorems/5ce8e666-53b7-4958-beed-be505dcf38ba?v=5bbf7092-58eb-4039-aad3-ddfbf22d02dd#clause-bilinear),

|G(x_{0}+z)-G(x_{0})-\langle q,z\rangle_{d}|=\bigl|-\bigl(F(x_{0}+z)-F(x_{0})-\langle DF(x_{0}),z\rangle_{d}\bigr)\bigr|\le\varepsilon\lVert z\rVert_{d},

by claim 2 of [Properties of the Absolute Value in an Ordered Field](/theorems/cbeef0cf-b6b0-4ce2-8b8a-d9fb43b5711a?v=1459a8b4-d373-4569-a686-e1567fc94db8). (ii) $G\circ\iota_{d}$ has a local minimum at $\lambda$ relative to $\mathcal{L}_{d}$: with $\delta>0$ from [Local Maximum of a Function Relative to a Subset of a Metric Space](/theorems/46837ef0-aa48-4e5b-a083-fce95fd065d7?v=67086c89-d85b-467e-86c7-6a79f8ecb396), every $\mu\in\mathcal{L}_{d}$ with $d_{\mathcal{L}}(\lambda,\mu)<\delta$ satisfies $F(\iota_{d}(\mu))\le F(\iota_{d}(\lambda))$, hence $G(\iota_{d}(\lambda))\le G(\iota_{d}(\mu))$ by claim 4 of [Elementary Order Arithmetic in an Ordered Field](/theorems/76f88645-0184-4f0e-a3c9-f9967fffeea7?v=64306ed0-32ee-43f6-a710-e12e0d00ce51). Steps 1 and 2 now give, for $v\in T^{+}_{\lambda}$, $0\le\langle-DF(x_{0}),v\rangle_{d}=-\langle DF(x_{0}),v\rangle_{d}$, that is $\langle DF(x_{0}),v\rangle_{d}\le0$ (claim 4 of [Elementary Order Arithmetic in an Ordered Field](/theorems/76f88645-0184-4f0e-a3c9-f9967fffeea7?v=64306ed0-32ee-43f6-a710-e12e0d00ce51)); and for $v\in T^{\pm}_{\lambda}$, $-\langle DF(x_{0}),v\rangle_{d}=0$, that is $\langle DF(x_{0}),v\rangle_{d}=0$.