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dWhen studying a mathematical space X (e.g. a vector space, a topological space, a manifold, a group, an algebraic variety etc.), there are two fundamentally basic ways to try to understand the space:

1. By looking at subobjects in X, or more generally maps $f: Y \to X$ from some other space Y into X.  For iTnstance, a point in a space X can be viewed as a map from $pt$ to X; a curve in a space X could be thought of as a map from ${}[0,1]$ to X; a group G can be studied via its subgroups K, and so forth.
2. By looking at objects on X, or more precisely maps $f: X \to Y$ from X into some other space Y.  For instance, one can study a topological space X via the real- or complex-valued continuous functions $f \in C(X)$ on X; one can study a group G via its quotient groups $\pi: G \to G/H$; one can study an algebraic variety V by studying the polynomials on V (and in particular, the ideal of polynomials that vanish identically on V); and so forth.

(There are also more sophisticated ways to study an object via its maps, e.g. by studying extensions, joinings, splittings, universal lifts, etc.  The general study of objects via the maps between them is formalised abstractly in modern mathematics as category theory, and is also closely related to homological algebra.)

A remarkable phenomenon in many areas of mathematics is that of (contravariant) duality: that the maps into and out of one type of mathematical object X can be naturally associated to the maps out of and into a dual object $X^*$ (note the reversal of arrows here!).  In some cases, the dual object $X^*$ looks quite different from the original object X.  (For instance, in Stone duality, discussed in Notes 4, X would be a Boolean algebra (or some other partially ordered set) and $X^*$ would be a compact totally disconnected Hausdorff space (or some other topological space).)   In other cases, most notably with Hilbert spaces as discussed in Notes 5, the dual object $X^*$ is essentially identical to X itself.

In these notes we discuss a third important case of duality, namely duality of normed vector spaces, which is of an intermediate nature to the previous two examples: the dual $X^*$ of a normed vector space turns out to be another normed vector space, but generally one which is not equivalent to X itself (except in the important special case when X is a Hilbert space, as mentioned above).  On the other hand, the double dual $(X^*)^*$ turns out to be closely related to X, and in several (but not all) important cases, is essentially identical to X.  One of the most important uses of dual spaces in functional analysis is that it allows one to define the transpose $T^*: Y^* \to X^*$ of a continuous linear operator $T: X \to Y$.

A fundamental tool in understanding duality of normed vector spaces will be the Hahn-Banach theorem, which is an indispensable tool for exploring the dual of a vector space.  (Indeed, without this theorem, it is not clear at all that the dual of a non-trivial normed vector space is non-trivial!)  Thus, we shall study this theorem in detail in these notes concurrently with our discussion of duality.