A topological vector space is a vector space equipped with a topology that makes the vector operations of addition and scalar multiplication continuous. This concept merges the algebraic structure of vector spaces with the topological structure, allowing for a deeper understanding of convergence and continuity in infinite-dimensional spaces. The interplay between these structures is essential in understanding important theorems and results in functional analysis, especially in areas such as the Closed Graph Theorem and the Banach-Alaoglu Theorem.
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