Daggers
Although \mathcal{H} and \mathcal{H}^\star are not identical spaces — the former is inhabited by kets, and the latter by bras — they are closely related.
There is a bijective map from one to the other given by |v\rangle\leftrightarrow \langle v|, and denoted by a dagger:
\begin{aligned}
\langle v|
&= (|v\rangle)^\dagger
\\|v\rangle
&= (\langle v|)^\dagger.
\end{aligned}
We usually omit the parentheses when it is obvious what the dagger operation applies to.
The dagger operation, also known as Hermitian conjugation, is antilinear:
\begin{aligned}
(c_1|v_1\rangle+c_2|v_2\rangle)^\dagger
&= c_1^\star\langle v_1| + c_2^\star\langle v_2|
\\(c_1\langle v_1|+c_2\langle v_2|)^\dagger
&= c_1^\star|v_1\rangle + c_2^\star|v_2\rangle.
\end{aligned}
Also, when applied twice, the dagger operation is the identity map.
You might already be familiar with Hermitian conjugation under another name: the conjugate transpose of an (n\times m) matrix A is an (m\times n) matrix A^\dagger, obtained by interchanging the rows and columns of A and taking complex conjugates of each entry in A, i.e. A^\dagger_{ij}=A^\star_{ji}.
In particular then,
|v\rangle = \begin{bmatrix}v_1\\v_2\\\vdots\\v_n\end{bmatrix}
\overset{\dagger}{\longleftrightarrow}
\langle v| = \begin{bmatrix}v_1^\star,&v_2^\star,&\ldots,&v_n^\star\end{bmatrix}.
We will come back to this \dagger operation on matrices in Section 0.6.