Remark

What this theorem tells us is that while studying spaces such as $L^{p}$ or $l_{p}$, we can use an inner-product like (but not an inner-product in the way we defined Hilbert spaces) expression to represent the set of all linear functions on $X$ by: $$\langle \cdot, y \rangle : X\owns x\mapsto \langle x, y \rangle =\int\limits _{\mathbb{R}}x(t)y(t) \, dt \in \mathbb{R}$$where $\langle \cdot, y \rangle$ is a continuous linear function on $X$, but this is equivalent to the function $y\in L^{q}(\mathbb{R}_{+};\mathbb{R})$ having an inner product-like expression with $x\in X$.

Remark

Likewise, for a discrete-time signal: $$\langle \cdot, y \rangle :X\owns x\mapsto \langle x, y \rangle =\sum_{i=1}^{\infty}x(i)y(i)\in \mathbb{R}$$is a linear function on $X$.

Remark

Thus, if $X=L^{p}(\mathbb{R}_{+};\mathbb{R})$ for $1\le p<\infty$, we can show that the dual space of $X$ is representable by elements in $L^{q}(\mathbb{R}_{+};\mathbb{R})$ where $\frac{1}{p}+\frac{1}{q}=1$.