I'm trying to understand why the following claim from my lecture notes is true:
Note that $\langle \psi | \hat{P} | \psi \rangle = 1 \iff \hat{P} \geq |\psi \rangle \langle \psi |$
where $| \psi \rangle$ refers to a unit vector in a separable Hilbert space $H$ and $\hat{P}$ is an orthogonal projection operator. I can see the equality but not the inequality (using the Dirac way of multiplying the expression by kets and bra).
I don't understand why is true. I'm new to functional analysis and most of the stuff I know is from its applications physics.
If $P$,$Q$ are orthogonal projectors, we say that $P\geq Q$ when $PQ = Q$. If $|\psi\rangle$ is a unit vector, you have : $$\langle \psi|P|\psi\rangle = \|P|\psi\rangle\|^2 \qquad \text{and}\qquad \|P|\psi\rangle\|^2 +\|(\mathbf 1-P)|\psi\rangle\|^2=1$$ and therefore : \begin{align} \langle\psi|P|\psi\rangle = 1 &\quad \Longleftrightarrow \quad (\mathbf 1-P)|\psi\rangle = 0 \\ &\quad \Longleftrightarrow \quad P|\psi\rangle = |\psi\rangle \\ &\quad \Longleftrightarrow \quad P |\psi\rangle\langle\psi| = |\psi\rangle\langle\psi| \\ &\quad \Longleftrightarrow \quad P\geq |\psi\rangle\langle\psi| \end{align}