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I'm a little bit confused about the definition of the weak topology. Brezis defines it as the coarsest topology which makes all the functions $\phi_f: E\to \mathbb{R}$ continuous. Where $E$ is a Banach space and $f\in E^*$, the dual of $E$. The function $\phi_f$ is defined as $\phi_f(x):=\langle f,x\rangle$. The $\langle \cdot,\cdot\rangle$ denotes a dual pairing. Are there more than one such dual pairing? If so, for every such pairing, the weak topology will be different for different pairings? So one has to specify the pairing when one talks about weak topologies?

Thanks in advance


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Different pairings will induce different topologies. So - strictly speaking - you have allways to keep in mind, which pairing is meant. For example on a dual Banach space $E^*$ you have the pairings \[ E \times E^* \to \mathbb K, \quad (x, f) \mapsto f(x) \] inducing the weak$^*$ topology and the pairing \[ E^{** } \times E^* \to \mathbb K, \quad (\phi, f) \mapsto \phi(f) \] inducing the weak topology.

In the Banach space context, the latter is usally adressed as the weak topology.

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