# Dual hilbert space

1. Mar 1, 2014

### wasi-uz-zaman

Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

2. Mar 1, 2014

### tiny-tim

hi wasi-uz-zaman!
it's not the same, it's back-to-front!

you can't add a ket to a bra

it's like ordinary 3D vectors and pseudovectors (a 3D pseudovector is a cross product of two 3D vectors)

you can't add a vector to a pseudovector …

they look as if they exist in the same space, but in fact the two spaces are back-to-front

3. Mar 1, 2014

### Staff: Mentor

Kets are elements of a vector space, bras are linear functions defined on the kets - entirely different things.

They are, with a few caveats such as Rigged Hilbert Spaces, isomorphic via the Rietz-Fisher Theorem - but that doesn't mean they are the same.

Thanks
Bill

4. Mar 2, 2014

### stevendaryl

Staff Emeritus
The easiest Hilbert space to deal with (or maybe the second easiest) is that of spin-1/2 states. Then the states are (or can be represented as) column matrices with 2 elements. The dual states are the row matrices with 2 elements. Obviously it doesn't make any sense to add a row and a column.