Positive definiteness refers to a property of a quadratic form or a matrix, where it ensures that all eigenvalues are positive, leading to a strictly positive value for all non-zero vectors. This concept is crucial in the study of normed linear spaces, as it guarantees that the associated norms derived from these quadratic forms behave nicely, such as being able to define a proper inner product. Understanding positive definiteness helps in analyzing convergence, stability, and optimization within these spaces.
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