Definitions for Manifolds, Measures and Hilbert Spaces
... the maps φ[U ∩ U 0 ] → Rn and φ0 [U ∩ U 0 ] → Rn are both smooth. An n-manifold, where n is a non-negative integer, is a set M , together with a collection C of n-charts on M , such that i) any two charts in the collection C are compatible, ii) the U ’s of the charts in C cover M , and iii) every n- ...
... the maps φ[U ∩ U 0 ] → Rn and φ0 [U ∩ U 0 ] → Rn are both smooth. An n-manifold, where n is a non-negative integer, is a set M , together with a collection C of n-charts on M , such that i) any two charts in the collection C are compatible, ii) the U ’s of the charts in C cover M , and iii) every n- ...
4 Images, Kernels, and Subspaces
... Definition. The image of a function consists of all the values the function assumes. If f : X → Y is a function from X to Y , then im(f ) = {f (x) : x ∈ X}. Notice that im(f ) is a subset of Y . Definition. The kernel of a function whose range is Rn consists of all the values in its domain at which ...
... Definition. The image of a function consists of all the values the function assumes. If f : X → Y is a function from X to Y , then im(f ) = {f (x) : x ∈ X}. Notice that im(f ) is a subset of Y . Definition. The kernel of a function whose range is Rn consists of all the values in its domain at which ...
Linear Algebra
... any other basis for X , i.e. either X is infinite dimensional or any basis has the same finite number of elements. Remark. Actually one can prove that, also for infinite dimensional vector spaces, two different sets of basis vectors always have the same cardinality. Proof. If the space is infinite d ...
... any other basis for X , i.e. either X is infinite dimensional or any basis has the same finite number of elements. Remark. Actually one can prove that, also for infinite dimensional vector spaces, two different sets of basis vectors always have the same cardinality. Proof. If the space is infinite d ...
Vector Spaces and Subspaces
... First fact : Every subspace contains the zero vector. The plane in R3 has to go through .0; 0; 0/. We mention this separately, for extra emphasis, but it follows directly from rule (ii). Choose c D 0, and the rule requires 0v to be in the subspace. Planes that don’t contain the origin fail those tes ...
... First fact : Every subspace contains the zero vector. The plane in R3 has to go through .0; 0; 0/. We mention this separately, for extra emphasis, but it follows directly from rule (ii). Choose c D 0, and the rule requires 0v to be in the subspace. Planes that don’t contain the origin fail those tes ...
