Positivity in a Hilbert Space

Question

Let $H$ be a Hilbert space with inner product $(\cdot,\cdot)$ and norm $\|\cdot \|$. Let $a,b,c\in H$ such that $a,b,c\neq 0$. Also $(b,c)=0$ i.e. they are orthogonal. Can we say $(a,b)+(a,c)+\|a\|^2>0$? Also if we drop the orthogonality then can we still prove the result?

Answer 1

Not true even if $b$ and$c$ are orthogonal. Take an orthonormal set $\{b,c\}=\{e_1,e_2\}$ and take $a=-\epsilon (b+c)$. Then the inequality fails whenever $0<\epsilon <1$.