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I'm reading Quantum Mechanics - A Modern Development, and it explains bra-ket notation, if I understand it correctly, as follows.

Let $V$ be a vector space, and let $F$ be a linear function mapping $V$ to a scalar, for $u,v \in V$, $F(au+bv)=aF(u)+bF(v)$

$F(v)=(f,v)$ is an inner product where $f$ is constructible by the Reisz Theorem.

$\langle F|v\rangle=F(v)$ by definition.

Later in the first chapter, there are theorems referring to things which look like this: $\langle \psi|\psi \rangle$ but, if the bras are linear functionals and kets are vectors, how is that a meaningful expression? How can something be both?

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Given an inner product $(\dot{},\dot{})$, $\langle x \rvert$ is the linear functional defined by $(\lvert x\rangle, \dot{})$.

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