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

In the setting of Real Hilbert Spaces: Standing Notation and Background, which puts the conventions of The Real Numbers: Standing Notation and Background in force, let $\mathbb{N}$ be the natural numbers, let $d\in\mathbb{N}$, let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples, and let the gauge space $E_{d}$ with its inner product $\langle\cdot,\cdot\rangle_{d}$ and norm $\lVert\cdot\rVert_{d}$ be as in The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge, and the embedding $\iota_{d}$ and the gauge distance $d_{\mathcal{L}}$ as in The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product, $E_{d}$ is a real inner product space with norm $\lVert\cdot\rVert_{d}$, and it is open in its own metric space, by Real Hilbert Spaces: Standing Notation and Background §topology; so Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space applies with $E=E_{d}$ and $U=E_{d}$. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space. Let $T^{+}_{\lambda}$ and $T^{\pm}_{\lambda}$ be the sets of inward and two-sided tangent vectors. Let $F:E_{d}\to\mathbb{R}$, let $\lambda\in\mathcal{L}_{d}$, and suppose that $F$ is differentiable at $\iota_{d}(\lambda)$, with gradient $DF(\iota_{d}(\lambda))$. Write $F\circ\iota_{d}$ for the map $\mu\mapsto F(\iota_{d}(\mu))$ on $\mathcal{L}_{d}$.

Then the following hold.

1. (Relative minimum)¶ If $F\circ\iota_{d}$ has a local minimum at $\lambda$ relative to $\mathcal{L}_{d}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$, then $0\le\langle DF(\iota_{d}(\lambda)),v\rangle_{d}$ for every $v\in T^{+}_{\lambda}$, and $\langle DF(\iota_{d}(\lambda)),v\rangle_{d}=0$ for every $v\in T^{\pm}_{\lambda}$.

2. (Relative maximum)¶ If $F\circ\iota_{d}$ has a local maximum at $\lambda$ relative to $\mathcal{L}_{d}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$, then $\langle DF(\iota_{d}(\lambda)),v\rangle_{d}\le0$ for every $v\in T^{+}_{\lambda}$, and $\langle DF(\iota_{d}(\lambda)),v\rangle_{d}=0$ for every $v\in T^{\pm}_{\lambda}$.