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A normed linear space $X$ is said to be smooth if for each $x \in X$ there exists a unique functional $x^* \in X^*$ with $\|x^*\|=1$ such that $x^*(x)=\|x\|$.

I know that $L_1[0,1]$ is not smooth. Hilbert spaces are smooth.

Which Banach spaces are smooth? In particular, does $\ell_{\infty}(I)$ smooth?

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The spaces $l^\infty$ and $L^\infty$ both are non-smooth.

The uniqueness of the functional does fail. Take $x\in l^\infty$ with $$ x=(1,1,0,\dots) $$ then $x_1^*$ and $x_2^*$ defined by $x_i^*(y)=y_i$ satisfy $x_1^*(x)=x_2^*(x)=1$.

Similarly for $L^\infty(0,1)$ take $x=1$ the constant function. Then any non-negative function in $L^1$ with mean value $1$ realizes the norm: $y\in L^1(0,1)$ with $\int y = 1$ and $y \ge 0$, then $$ y(x):=\int_0^1 yx = 1 = \|x\|_{L^\infty}. $$

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Smooth spaces are, e.g., $L^p(0,1)$ for $1 < p < \infty$.

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