All continuous bounded linear functionals have spaces {dual space} {adjoint space} {Banach space}. Dual-space norms are functional bounds. Duals change L^q to L^p, where q = p / (p - 1). Dual spaces are generalized Hilbert spaces.
Spaces {complex sequence space} can be equivalent to L^2 spaces of complex-valued, measurable, and square integrable functions. Complex space has inequalities {Schwarz's inequality} {Parseval's inequality}.
Spaces {function space} can have functions as points or distances. Spaces can have finite numbers of elements. Spaces can have infinite numbers of elements, with two limiting elements at interval ends {closed set, function}. Spaces can have no sequence gaps {sequentially compact}. Spaces can have only one limit element {relatively sequentially compact}.
Function spaces can use generalized Pythagorean theorem.
Function spaces can have triangle inequality, Schwarz's inequality, and other inequalities {Bessel's inequality}.
Mutually orthogonal space elements are linearly independent.
Complex abstract spaces {Hilbert space, analysis} can have infinite dimensions.
transformation
Coordinates can transform. Integral from x = a to x = b of K(x,y) * u(x) * dx can transform differentiable function u(x), where K(x,y) are differential equations.
purposes
Differential and integral equation eigenvalue theory is similar to n-dimensional-space linear transformations. Quantum-mechanics equations can use Hilbert-space spectral theory. Observed values are linear symmetric operators in Hilbert space. Linear symmetric energy-operator eigenvalues and eigenfunctions are energy levels. Eigenvalue differences show emitted-light frequencies. Lebesgue integrable functions {square summable function} make spaces similar to Hilbert spaces of sequences. Banach spaces {complete normed vector space} are generalized Hilbert spaces.
Spaces {linear vector space} can have complete orthonormal countable sets. Topology is about equivalencies during continuous motions. Linear vector space can define metric using the norm {strong topology}. Linear vector spaces can have systems of neighborhoods with weak convergence {weak topology}.
Banach space includes spaces {L^p space}, of continuous bounded measurable functions, which does not need two orthogonal elements and in which inner product does not define norm.
Spaces {metric space} can have distances. Integral from x = a to x = b of K(x,y) * u(x) * dx, where K(x,y) are differential equations, can transform distance function u(x) into another distance function.
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Date Modified: 2022.0225