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Serge Lang's Linear Algebra textbook just introduced me to the concept of dual space in very formal terms: space of all functional transformations having co-domain as $1$-dimensional vector space over the field $\mathbb{K}$ (since in essence, field $\mathbb{K}$ is a vector space over itself).

But the textbook did not explain the exact purpose of the term "duality", thus I decided to go little further and dive into some basic functional analysis.

The Uncertainty Principle by Terrence Tao (reference):

Terrence Tao wrote a really nice article on the concept of duality, that is explained in terms of local and global perspectives. In his first example:

Vector space duality A vector space ${V}$ over a field ${F}$ can be described either by the set of vectors inside ${V}$, or dually by the set of linear functionals ${\lambda: V \rightarrow F}$ from ${V}$ to the field ${F}$ (or equivalently, the set of vectors inside the dual space ${V^*}$). (If one is working in the category of topological vector spaces, one would work instead with continuous linear functionals; and so forth.) A fundamental connection between the two is given by the Hahn-Banach theorem (and its relatives).

As you see in the last sentence (in italic font), Tao mentions that Hahn-Banach theorem displays the fundamental connection between some vector space $V$ and its dual $V^*$. Therefore I've decided to investigate this concept a little further.

Hahn-Banach Theorem and Dual space:

There is a question regarding some similar connection on Math SE, but I'm not certain whether or not it is the answer to my question.

From my understanding of answers below the referenced question, Hahn-Banach theorem states that for any arbitrary vector $v \in V$, there exists a functional $L \in V^*$ such that $|L(v)|=||v||_{V}$ and $||L||_{V^*}=1$.

The definition of norm on the dual space is:

$$||L||_{V^*}=\textrm{sup}\{|L(v)|: v \in V, |v| \leq 1 \}$$

where $\textrm{sup}$ denotes the supremum of set.

I also know that every $L \in V^*$ is a linear transformation with norm $1$ that is bounded (i.e $\exists C \in \mathbb{K}, ||T(v)||_{V^*} < C||v||_{V}, \forall v \in V$, where $C$ is called operator norm). This (along with definition of dual norm) shows another interesting relation:

$$||v||_{V}=\textrm{sup}\{|L(v)|: v \in V, ||L||_{V^*}=1 \}$$

Riesz-representation theorem (Extension):

According to comments made by Berci below this post, complete inner product spaces (or Hilbert spaces) have special relationship with their dual spaces. Let $H$ be a Hilbert Space on the field $\mathbb{R}$, this relationship can be seen by Riesz-representation theorem which asserts that $H$ and $H^*$ are isometrically isomorphic (whereas in complex field case, they are anti-isomorphic).

In more specific details, it shows that there exists $g \in H$ such that for any functional $L \in H^*$ and any $x \in H$: $L(x) = \langle{} f, g \rangle{}$. Moreover, as a consequence to isometric connection: $||x||_{H} = ||L(x)||_{H^*}$.

This theorem establishes interesting connection between inner product and functionals. In fact, I believe it can be utilized as extension for Hahn-Banach theorem to see the deeper connection from geometric perspective, since the isomorphic isometric connection that Riesz Representation gives is equivalent of hyper-plane corresponding to its normal unit vector (and this seems to be a consequence of Hahn-Banach theorem, proving the existence of unit functionals). This can be more intuitively understood by specific cases of $L^P$ spaces, since they have interesting properties such as natural isomorphism of their duals, but I don't believe I have sufficient experience to group this information yet.


Question:

How exactly does the Hahn-Banach theorem show the fundamental connection between a vector space and its dual, as mentioned by Terrence Tao? Is it just that every vector $v$ has a corresponding functional which has the norm $||v||$? Is there more abstract explanation involving the idea of dual norm?

Thank you!

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    $\begingroup$ Note that functionals with norm 1 correspond to hyperplanes. Moreover, in Hilbert spaces, where orthogonality makes sense, the Riesz representation gives a one-to-one correspondence between vectors and functionals: geometrically, a hyperplane then corresponds to its unit normal vector.. $\endgroup$ – Berci May 27 at 22:47
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    $\begingroup$ The comment of Berci is actually a good answer. Anyway, I remember that I got a much better grasp of these things by studying the case of $L^p$ spaces, in which many things are explicit: see, for example, this question. $\endgroup$ – Giuseppe Negro May 28 at 9:48
  • $\begingroup$ @Berci Hello, thank you for your comment. So Riesz represesantation theorem basically shows that every functional in dual space (of complete inner product space) can be written as inner product. But I'm unable to understand the concept of hyperplanes in this context, is hyperplane specific subspace of V in this case (that is mapped to dual space through injective transformation)? Sorry for misunderstanding. $\endgroup$ – ShellRox May 28 at 10:28
  • $\begingroup$ @Giuseppe Negro Thank you for the reference, this would definitely help for understanding the main idea of Riesz representation theorem. $\endgroup$ – ShellRox May 28 at 10:40
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    $\begingroup$ For a given functional, if you colour the points of a (topological) vector space according to values, you get colored strips of parallel affine hyperplanes. If there's an inner product, we can identify this with a (well specified) normal vector of these hyperplanes. $\endgroup$ – Berci May 28 at 13:23

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