Vanishing of Uniformly Small Linear, Quadratic and Bilinear Terms in a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E$ be a real inner product space, with its inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and zero vector $0_{E}$ as fixed there, and let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, with the norm $\lVert\cdot\rVert$ and the zero form $0_{\mathrm{Sym}}$ fixed there.

Let $p\in E$ and let $b\in\mathrm{Sym}(E)$. Then the following hold.

1. (A uniformly small linear term vanishes)¶ Suppose that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that every $z\in E$ with $|z|<\delta$ satisfies

Then $p=0_{E}$.

2. (A uniformly small quadratic term vanishes)¶ Suppose that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that every $z\in E$ with $|z|<\delta$ satisfies

Then $b=0_{\mathrm{Sym}}$.

3. (A uniformly small bilinear term vanishes)¶ Suppose that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that all $w,y\in E$ with $|w|<\delta$ satisfy

Then $b=0_{\mathrm{Sym}}$.

4. (A linear and a quadratic term together)¶ Suppose that for every positive $\varepsilon\in\mathbb{R}$ there is a positive $\delta\in\mathbb{R}$ such that every $z\in E$ with $|z|<\delta$ satisfies

Then $p=0_{E}$ and $b=0_{\mathrm{Sym}}$.