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:
$$||v||_{V}=\textrm{sup}\{|L(v)|: v \in V, ||L||_{V^*}=1 \}$$
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!
from Hot Weekly Questions - Mathematics Stack Exchange
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