This article, presented by A. Grothendieck at the Séminaire N. Bourbaki in 1954, delves into the theory of topological tensor products and nuclear spaces. The author discusses the definitions and properties of topological tensor products, focusing on the norms and topologies induced by these products. Key topics include: 1. **Topological Tensor Products**: - Definitions and existence of norms for topological tensor products \( E \otimes F \) of Banach spaces \( E \) and \( F \). - The completion of \( E \otimes F \) with respect to these norms, denoted \( E \otimes^{\overline{}} F \). - The dual of \( E \otimes F \) is identified with \( B(E, F) \), the space of continuous bilinear forms on \( E \times F \). 2. **Nuclear Spaces**: - Definitions and characterizations of nuclear spaces, including the condition that \( E \otimes F \cong E \otimes^{\overline{}} F \) for all locally convex spaces \( E \) and \( F \). - Equivalence conditions for a space \( E \) to be nuclear, such as the weak compactness of sequences in \( E \). - Properties of nuclear spaces, including the density of \( E \otimes F \) in \( E \otimes^{\overline{}} F \) when \( E \) and \( F \) are complete. 3. **Applications and Examples**: - Applications of topological tensor products in various contexts, such as differential operators and holomorphic functions. - Examples of nuclear spaces, including spaces of smooth functions on differentiable manifolds and holomorphic functions on complex manifolds. 4. **Properties of Fredholm Operators**: - The representation of Fredholm operators in terms of topological tensor products. - The decay rate of eigenvalues for Fredholm operators, which is often rapid. The article provides a comprehensive overview of the theory, including detailed proofs and examples, making it a foundational reference in functional analysis.This article, presented by A. Grothendieck at the Séminaire N. Bourbaki in 1954, delves into the theory of topological tensor products and nuclear spaces. The author discusses the definitions and properties of topological tensor products, focusing on the norms and topologies induced by these products. Key topics include: 1. **Topological Tensor Products**: - Definitions and existence of norms for topological tensor products \( E \otimes F \) of Banach spaces \( E \) and \( F \). - The completion of \( E \otimes F \) with respect to these norms, denoted \( E \otimes^{\overline{}} F \). - The dual of \( E \otimes F \) is identified with \( B(E, F) \), the space of continuous bilinear forms on \( E \times F \). 2. **Nuclear Spaces**: - Definitions and characterizations of nuclear spaces, including the condition that \( E \otimes F \cong E \otimes^{\overline{}} F \) for all locally convex spaces \( E \) and \( F \). - Equivalence conditions for a space \( E \) to be nuclear, such as the weak compactness of sequences in \( E \). - Properties of nuclear spaces, including the density of \( E \otimes F \) in \( E \otimes^{\overline{}} F \) when \( E \) and \( F \) are complete. 3. **Applications and Examples**: - Applications of topological tensor products in various contexts, such as differential operators and holomorphic functions. - Examples of nuclear spaces, including spaces of smooth functions on differentiable manifolds and holomorphic functions on complex manifolds. 4. **Properties of Fredholm Operators**: - The representation of Fredholm operators in terms of topological tensor products. - The decay rate of eigenvalues for Fredholm operators, which is often rapid. The article provides a comprehensive overview of the theory, including detailed proofs and examples, making it a foundational reference in functional analysis.