Dual Space of Lᵖ

• categories: Functional analysis, Theorem

The Dual Space $(L^p([a,b]){)}^{\ast }$ consists of all continuous linear functionals on $L^p([a,b])$, where $1\le p<\infty$.

• For $1<p<\infty$, the dual space $(L^p{)}^{\ast }$ is isometrically isomorphic to $L^q([a,b])$, where $\frac{1}{p}+\frac{1}{q}=1$. Specifically, for $f\in L^q([a,b])$, the corresponding functional $T_f$ is defined by:

  $$T_f(g)={\int }_a^{b}f(x)g(x)dx,\text{for all }g\in L^p([a,b]).$$

• For $p=1$, the dual space $(L^1{)}^{\ast }$ is $L^{\infty }([a,b])$, with the same type of pairing:

  $$T_f(g)={\int }_a^{b}f(x)g(x)dx,\text{for all }g\in L^1([a,b]).$$

• Properties:

  • The functional $T_f$ is bounded and linear.
  • The norm $\Vert T_f\Vert$ in the dual space satisfies $\Vert T_f\Vert =\Vert f{\Vert }_q$, establishing an isometric isomorphism.

• Intuition:

  • The dual pairing $⟨f,g⟩=\int fgdx$ connects the function spaces $L^p$ and $L^q$, where they act as ‘partners’ under the Hölder’s Inequality, which ensures that such pairings are finite.