In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general...
I believe its like this...let me first recall bits of that proof. The inequality $\|Tf\|_{q_\theta} \le M_0^{1-\theta} M_1^\theta \|f\|_{p_\theta}$ is equivalent to
$$ \sup_{\substack{\|f\|_{p_\theta}=1\\ \|g\|_{q_\theta'}=1}
} \int_Y (T f)g \le M_0^{1-\theta} M_1^\theta $$
Where the sup...
