Uniqueness of bra-ket correspondence

Two complex topological vector spaces $X$ and $Y$ are said to be in duality if there is a sesquilinear map $$b:X\times Y\to \mathbb{C}\; .$$

The idea is that, given such map, a dual action of $X$ on $Y$ and viceversa is defined: for any $x\in X$, $$b(x,\cdot): Y\to \mathbb{C}$$ is a linear functional.

There are two natural spaces in duality with a given topological vector space $X$. One is the space of all linear functionals, called the algebraic dual $X^*$; the other is the space of all continuous (w.r.t. the topology on $X$) functionals, called the continuous dual $X'$.

The continuous dual has some special properties, and it is often easy to characterize explicitly, given the space $X$. For example, if $X$ is a Banach space, then also on the dual there is a complete norm "inherited" from the one on the original space. The Lebesgue space $L^p$, for example, has the Lebesgue space $L^{\frac{p}{p-1}}$ as continuous dual, provided $1\leq p<\infty$.

For Hilbert spaces $\mathscr{H}$, the continuous dual has an even nicer property: it is the space itself. In fact, the inner product is a sesquilinear form, that makes the Hilbert space in duality with itself. In addition, given $\psi\in \mathscr{H}$, $$\langle \psi,\cdot\rangle: \mathscr{H}\to \mathbb{C}$$ is continuous. Therefore, $\mathscr{H}\subseteq \mathscr{H}'$ (with the suitable identification of the ket $\psi$ with the bra $\langle \psi, \cdot\rangle$).

On the other hand, as proved in the Riesz's representation theorem cited in the comments, the converse is also true: $\mathscr{H}'\subseteq \mathscr{H}$, that is every continuous linear functional can be written as the bra of a given vector in the space.

Therefore, $\mathscr{H}=\mathscr{H}'$. The algebraic dual, on the other hand, is bigger and there are thus functionals that are not continuous and cannot be written as the bra of an element in the Hilbert space.

This is actually one of the instances where bra-ket notation is a bit confusing, because the exact nature of the duality map is built into the notation.

So instead, let us just denote by $\psi$ an element of your (complex) Hilbert space $H$. We can define the dual of $H$ as $H^*:=\text{Hom}(H,\mathbb{C})$, which is mathematician-speak for “the space of linear transformations $f:H\to\mathbb{C}$.” One can show, under certain technical assumptions physicists don’t worry about, that $H$ and $H^*$ are isomorphic as complex Hilbert spaces. (As pointed out by yuggib, this fails to be true when the Hilbert space is infinite-dimensional.) However, just as you noted, this isomorphism isn’t unique.

There is a particularly nice choice of isomorphism, though. Recall that Hilbert spaces are equipped with a complete Hermitian inner-product, which we will denote $\langle\cdot,\cdot\rangle$. That is, given two vectors $\psi$ and $\varphi$, then $\langle\psi,\varphi\rangle$ is their inner-product (this is already very similar to bra-ket notation). It turns out that the inner-product gives us a natural isomorphism between $H$ and $H^*$. We associate to every $\psi\in H$ a corresponding $\psi^*\in H^*$ which acts on $H$ as

$$\psi^*(\varphi)=\langle\psi,\varphi\rangle$$

for all $\varphi$ in $H$. This uniquely defines the desired isomorphism.

This is already built into bra-let notation, where we would denote $\psi$ as $|\psi\rangle$ and $\psi^*$ as $\langle\psi|$. In this notation, the definition of $\langle\psi|$ is that it acts as

$$\langle\psi|(|\varphi\rangle)=\langle\psi,\varphi\rangle,$$

which we might as well write as $\langle\psi|\varphi\rangle$. The fact that the inner-product-respecting ismomorphism is naturally built in is really where the power of bra-ket notation comes from, but is a detail that is really often skipped over in undergraduate courses, opting for the “complex conjugate transpose”, basis-dependent definition.

In the words of Fredric Schuller:

A gentleman only chooses a basis when he must.

Of course if you just define it as you said then you can have many such functionals which map this ket to real numbers. That is why we should define such functionals which when acting on a basis give ones and zeroes. That would be the basis for the space of functionals. Such space is called dual and the basis is dual basis of the dual space. Now you can prove that dual of this dual is Just the original space basis, the one on which functionals were defined to act. I hope this helps a bit.