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)(i) 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

$$|\langle p,z\rangle|\le\varepsilon\,|z| .$$

Then $p=0_{E}$.

2. (A uniformly small quadratic term vanishes)(ii) 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

$$|b(z,z)|\le\varepsilon\,|z|^{2} .$$

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

3. (A uniformly small bilinear term vanishes)(iii) 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

$$|b(w,y)|\le\varepsilon\,|w|\,|y| .$$

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

4. (A linear and a quadratic term together)(iv) 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

$$\Bigl|\langle p,z\rangle+\tfrac{1}{2}\,b(z,z)\Bigr|\le\varepsilon\,|z|^{2} .$$

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