# What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my textbook (Griffiths) and read countless of pdf’s on the web but for some reason I am just not getting it.

Can someone explain to me, in the simplest terms possible, what this “Bra” and “Ket” (Dirac) notation is, why it is so important in quantum mechanics and how it relates to Hilbert spaces? I would be infinitely grateful for an explanation that would actually help me understand this.

Edit 1: I want to thank everyone for the amazing answers I have received so far. Unfortunately I am still on the road and unable to properly read some of the replies on my phone. When I get home I will read and respond to all of the replies and accept an answer.

Due to the Riesz-Frechet theorem, a correspondence can be established between $\mathcal{H}$ and the space of linear functionals where the bras live, thereby the maybe slightly ambiguous notation.