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The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and

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• The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and (en)
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• Let and be Hilbert spaces and endow with the trace-norm. When the space of compact linear operators is equipped with the operator norm then its dual is and its bidual is the space of all continuous linear operators (en)
• Let and be metrizable locally convex TVSs and let Then is the sum of an absolutely convergent series where and and are null sequences in and respectively. (en)
• The canonical embedding becomes an embedding of topological vector spaces when is given the projective topology and furthermore, its range is dense in its codomain. If is a completion of then the continuous extension of this embedding is an isomorphism of TVSs. So in particular, if is complete then is canonically isomorphic to (en)
• Let and be locally convex TVSs with nuclear. Assume that both and are Fréchet spaces or else that they are both DF-spaces. Then: # The strong dual of can be identified with ; # The bidual of can be identified with ; # If in addition is reflexive then is a reflexive space; # Every separately continuous bilinear form on is continuous; # The strong dual of can be identified with so in particular if is reflexive then so is (en)
• Let and be Fréchet spaces and let be a balanced open neighborhood of the origin in . Let be a compact subset of the convex balanced hull of There exists a compact subset of the unit ball in and sequences and contained in and respectively, converging to the origin such that for every there exists some such that (en)
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• Theorem (en)
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• Grothendieck (en)
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• The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and (en)
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• Projective tensor product (en)
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