Deriving identity $\|\langle\phi_1|\phi_1 \rangle + \langle\phi_2|\phi_2 \rangle\| \leq \| \space|\phi_1 \rangle \space\| + \| |\phi_2 \rangle \|$

Question

Consider two arbitrary vectors $|\phi_1 \rangle$ and $| \phi_2 \rangle$ belonging to the inner product space:

My notes then state the following inequality holds but I have no idea where it came from:

$$ \|\space|\phi_1 \rangle + \space|\phi_2 \rangle \| \leq \| \space|\phi_1 \rangle \space\| + \|\space |\phi_2 \rangle \space \|$$

Any tips on how to understand where this come from would be appreciated.

Answer 1

According to the Cauchy-Schwarz inequality, it results that: \begin{align*} (\|v\| + \|w\|)^{2} & = \|v\|^{2} + 2\|v\|\|w\| + \|w\|^{2}\\\\ & \geq \|v\|^{2} + 2|\langle v,w\rangle| + \|w\|^{2}\\\\ & \geq \|v\|^{2} + 2\text{Re}\langle v,w\rangle + \|w\|^{2}\\\\ & = \|v + w\|^{2} \end{align*}

and we are done.

Hopefully this helps!