There are two wolfs inside you...

Consider a vector space $X$ over a field $\mathbb{K}$ (usually^[1] $\mathbb{R}$ or $\mathbb{C}$), endowed with a topology $\tau$ that is compatible with its vector space structure –that is, sum, and scalar multiplication are continuous operations.

In principle, we can consider the set of all linear maps from $X$ to the field $\mathbb{K}$. This set is called the algebraic dual of $X$ and is denoted by $X^* = \{ f:X\to\mathbb{K} \mid f \text{ is linear} \}.$ Note that this defines a vector space over $\mathbb{K}$, which is not endowed with any topology; it is, in principle, just an algebraic construct.

If we restrict ourselves to the set of continuous linear maps, we obtain the topological dual of $X$, denoted by $X' = \{ f:X\to\mathbb{K} \mid f \text{ is continuous w.r.t. } \tau, \; f \text{ is linear} \}.$ This is a vector space over $\mathbb{K}$, and even if in principle it is not endowed with a particular topology –thus, it is not a topological vector space–, it can be endowed with different topologies^[2].

...and they seem to like each other...

At times, it is more convenient –as in theory of distributions– to put different spaces in duality with each other. In this case, one considers a (canonical) pairing $\langle \cdot, \cdot \rangle$ between two spaces $X$ and $Y$, which is a bilinear map in $X \times Y$. The main requirement is a separation condition (or injectivity condition, or non-degeneracy condition): if $\langle x, y \rangle = 0$ for all $y\in Y$, then $x=0$; similarly, if $\langle x, y \rangle = 0$ for all $x\in X$, then $y=0$.

...but let's suppose they are nice...

The above were a glimpse of the duality theory, a very interesting topic treated with a lot of care in the literature.

Following ??, we restrict ourselves to the case of Banach spaces. A Banach space is a complete normed vector space, that is, a vector space $X$ over a field $\mathbb{K}$, endowed with a norm $\|\cdot\|$.

Despite the simplification, one can still consider a quite important question: what happens if we keep taking biduals of a Banach space $X$? Note that $X \hookrightarrow X''$, where $X''$ is the bidual of $X$, in a natural way, given by the canonical embedding

$$\iota:X \to X'', \quad \iota(x)(f) = f(x), \text{ for all } f\in X'.$$

?? studies this questions, and define the two natural categorical limits, making use of the standard constructions from TVS theory.

(TODO)

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[1]	Can we consider more general settings? Usually yes.

For instance, see ?? and ??. Concerning LCTVSs, these fields suffice.

[2]	Some cases are worth mentioning: the weak topology, the Mackey topology,

and the strong topology. The terminology is not great, and it can be confusing at times... These constructions can be thought in terms of their generating basis and/or seminorms.