My quantum mechanics teacher told us during the class that there were many more "bra" than "kets", but I confess that I don't quite understand this. Indeed, in quantum mechanics, we work in the space of square integrable function $L^2(\mathbb{R}^3)$ such that each ket can be associated with a wave function $\varphi \in L^2(\mathbb{R}^3)$. We have defined an bra as a linear form on $L^2(\mathbb{R}^3)$, i.e. such that each bra can be associated to an $\ell \in (L^2(\mathbb{R}^3))'$ (the dual of $L^2(\mathbb{R}^3)$). By Riesz's representation theorem, there is an unique function $\psi \in L^2(\mathbb{R})$ such that $$\ell(\varphi) = \int_\mathbb{R} \psi^*\varphi,\quad \forall \varphi \in L^2(\mathbb{R}^3).$$ This shows that there is an isomorphism between $L^2(\mathbb{R}^3)$ and its dual. How can there be more bra than ket? Thanks a lot!

My quantum mechanics teacher told us during the class that there were many more "bra" than "kets", but I confess that I don't quite understand this. Indeed, in quantum mechanics, we work in the space of square integrable function $L^2(\mathbb{R}^3)$ such that each ket can be associated with a wave function $\varphi \in L^2(\mathbb{R}^3)$. We have defined an bra as a linear form on $L^2(\mathbb{R}^3)$, i.e. such that each bra can be associated to an $\ell \in (L^2(\mathbb{R}^3))'$ (the dual of $L^2(\mathbb{R}^3)$). By Riesz's representation theorem, there is an unique function $\psi \in L^2(\mathbb{R})$ such that $$\ell(\varphi) = \int_\mathbb{R} \psi^*\varphi,\quad \forall \varphi \in L^2(\mathbb{R}^3).$$ This shows that there is an isomorphism between $L^2(\mathbb{R}^3)$ and its dual. How can there be more bra than ket?