../

Dual Norm

$\newcommand{\K}{\mathbb{K}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\norm}{||\cdot||~}$ $\newcommand{\hp}{\quad\quad\square}$

Let $X$ be some normed vector space over the field $\K$ with norm

$\norm:X \rightarrow \K$

Then, we define the set $X’$ to the set of all linear functionals

$X’ = \{l:X\rightarrow \K\ | l\text{ is a linear functional}\}$

This set is called the dual space of $X$.

Theorem 1: $X’$ is also a vector space

We define an operation $+: X’ \times X’ \rightarrow X’$ as follows

$(f + g)(x) = f(x) + g(x)$

$(f + g)$ takes in $x\in X$ and outputs a $\K$ so it is a valid functional

Does it satisfy the linear properties?

$(f+g)(\alpha x) = f(\alpha x) + g(\alpha x) = \alpha f(x) + \alpha g(x) = \alpha (f(x)+g(x)) = \alpha (f+g)(x)\hp$

$(f+g)(x+y) = f(x+y) + g(x+y) = f(x)+f(y)+g(x)+g(y)$

$(f+g)(x) + (f+g)(y) = f(x)+g(x)+f(y)+g(y)=f(x)+f(y)+g(x)+g(y)\hp$

We define the scalar multiplication $*: X’ \times \K \rightarrow X’$ as follows

$(aV)(x) = a(V(x))$

It is clear that $a(V(x)) \in X’$ as it takes an $x\in X$ and performs $V(x)$ and multiplies the result with $a$ ending up in $\K$. So it is a valid functional.

$(aV)(\alpha x) = a(V(\alpha x)) = a\alpha V(x) = \alpha(aV(x)) = \alpha(aV(x))\hp$

$(aV)(x+y) = a(V(x+y)) = a(V(x) + V(y)) = a(V(x)) + a(V(y)) = (aV)(x) + (aV)(y)\hp$

So it is linear

Does it satisfy the 8 axioms of a vector space?

  1. Associativity of vector addition

$(a + (b + c))(x) = ((a + b) + c)(x)$

$a(x) + (b+c)(x) = (a+b)(x) + c(x)$

$a(x) + b(x) + c(x) = a(x) + b(x) + c(x)\hp$

  1. Identity of vector addition

We set the identity to the linear functional zero $x \mapsto 0, \forall x \in X$

$(f + 0)(x) = f(x) + 0(x) = f(x) + 0 = f(x)$

$(0 + f)(x) = 0(x) + f(x) = 0 + 0(x) = f(x)\hp$

  1. Inverse of vector addition

We set the inverse to be the negation

$f^{-1}(x) = -f(x) = f(-x)$ which always exists

$f(x) + f^{-1}(x) = f^{-1}(x) + f(x) = -f(x) + f(x) = 0$

  1. Commutativeness of vector addition

$(f + g)(x) = (g + f)(x)$

$f(x) + g(x) = g(x) + f(x)$. This is addition in the field $\K$ which is commutative.

  1. Compatibility of scalar multiplication with field multiplication

$(ab(V))(x) = (a(bV))(x)$ Definition of scalar multiplication

$(ab)(V(x)) = a(b(V(x)))$ Field multiplication is associative

$abV(x) = abV(x)\hp$

  1. Identity of scalar multiplication

$(1V)(x) = 1V(x) = V(x)\hp$

  1. Distributiveness of scalar multiplication w.r.t to vector addition

$(a(V + W))(x) = a((V+W)(x)) = a(V(x) + W(x)) = (aV)(x) + (aW)(x) \hp$

  1. Distributiveness of scalar multiplication w.r.t to field addition

$((a + b)V)(x) = (a+b)(V(x)) = aV(x) + bV(x) = (aV)(x) + (bV)(x) \hp$

Norm

We define the norm of the vector space as

$||z||_* = \sup \{ z(x) : ||x|| \le 1\}$

Refer 20260904T095633-hw1 for the proof