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Notions of vector calculus
Abstract vector spaces
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Banach space belonging bounded called Cauchy sequence Cauchy-Schwarz inequality classes of equivalence clearly closed set closed subspace closure complex numbers complex space condition Consequently consider contains continuous functions continuous transformation converge strongly convex coordinates corresponding bilinear decreasing sequence define denote determined dimension direct sum distance domain exists fact finite number formula functional F given gives graph Hilbert space Hull Hull2 implies infinite sequences intersection interval 0;l inverse isomorphism limit linear combination linear transformation linearly independent linearly independent vectors means metric space normed vector space notion obtain one-dimensional one-to-one correspondence open sets orthogonal sum plane positive definite proof proper norm prove pseudo-norm quadratic form F(u quotient space real numbers real space reduced angle representative subspace satisfies scalar product sequential space strong convergence subsequence subset unitary space vector f weak convergence zero element