= Solution
The <continuous dual space> $X'$ consists of all bounded linear maps $\Lambda:X\to\mathbb C$, with
$$
\|\Lambda\|_{X'}
=\sup_{\|x\|_X\leq1}|\Lambda x|.
$$
If $Y=L^p(\mathbb R^n)$ with $1\leq p<\infty$ and $q$ is the <Holder inequality>[conjugate exponent], then <duality of Lp spaces> identifies $Y'$ isometrically with $L^q(\mathbb R^n)$, including $q=\infty$ when $p=1$. Every functional has the form
$$
\Lambda_h(f)=\int_{\mathbb R^n}f(x)\overline{h(x)}\,dx,
\qquad h\in L^q,
$$
and $\|\Lambda_h\|=\|h\|_q$.
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