Dual Space

Definition

$X$ be a normed vector space $X^{\ast }=\{l:X\to F\text{linear},\Vert l\Vert ={\sup }_{x\in X,\Vert x\Vert =1}|l(x)|<\infty \}$ is the dual space of $X$ and it’s a normed vector space with $\Vert \cdot \Vert$.

$(X^{\ast },\Vert \cdot \Vert )$ is complete if the base field is $\mathbb{R}\text{or}\mathbb{C}$

Proof This is a direct result from a more general theorem.

Remark

In general we won’t achieve supremum in $\Vert L{\Vert }_{X^{\ast }}$. Here’s an example $X=\{f\in C[-1,1]:f(0)=0\}$ and $L(f)={\int }_{-1}^{1}f(x)dx$.