Articles of dual spaces

Separability of Banach Spaces

A homework problem from Folland Chapter 5, problem 5.25. If $\mathcal{X}$ is a Banach space and $\mathcal{X}^{\star}$ is separable, then $\mathcal{X}$ is separable. I tried the following approach: For every $\epsilon >0$ I wanted to show the existence of a linear map from $x_{1},\ldots,x_{n}$ such that for any $x\in\mathcal{X}$ $\| x-L(x_{1},\ldots,x_{n})\|\leq \epsilon$.

Dual of $l^\infty$ is not $l^1$

I know that the dual space of $l^\infty$ is not $l^1$, but I didn’t understand the reason. Could you give me a example of an $x \in l^1$ such that if $y \in l^\infty$, then $ f_x(y) = \sum_{k=1}^{\infty} x_ky_k$ is not a linear bounded functional on $l^\infty$, or maybe an example of a $x […]

The Duals of $l^\infty$ and $L^{\infty}$

Can we identify the dual space of $l^\infty$ with another “natural space”?. If the answer yes what about $L^\infty$. By the dual space I mean the space of all continuous linear functionals.

Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don’t really understand why we want to study them within linear algebra. I was wondering if anyone knew a nice intuitive motivation for the study of dual […]

Why are vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the vector space $V=\mathbb F^{(\kappa)}$ (that is an infinitely dimensional space with basis of cardinality $\kappa$) is not isomorphic (as […]

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I’ve read in several places that one motivation for category theory was to be able to give precise meaning to statements like, “finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as well, but not canonically.” I’ve finally sat down to work through this, and – Okay, […]

Prove that $X^\ast$ separable implies $X$ separable

Can someone tell me if I got the following right: Assume $X$ to be a normed vector space over $\mathbb{R}$. Prove that if the dual space $X^\ast$ is separable then $X$ is separable as well. I’m supposed to use the following hint: First show that for each $x_n^\ast$ we may choose a unit vector $x_n […]