Fall 2015
... It follows that |f (xni ) − f (z)| and |f (yni ) − f (z)| are less than ϵ1 , producing a contradiction ϵ0 ≤ |f (xni ) − f (yni )| ≤ |f (xni ) − f (z)| + |f (z) − f (yni )| < ϵ1 + ϵ1 = ϵ0 to the assumption that f is not uniformly continuous. T2. Suppose X is a Hausdorff space which has no isolated poi ...
... It follows that |f (xni ) − f (z)| and |f (yni ) − f (z)| are less than ϵ1 , producing a contradiction ϵ0 ≤ |f (xni ) − f (yni )| ≤ |f (xni ) − f (z)| + |f (z) − f (yni )| < ϵ1 + ϵ1 = ϵ0 to the assumption that f is not uniformly continuous. T2. Suppose X is a Hausdorff space which has no isolated poi ...
CORE 4 Summary Notes
... SEPARATING THE VARIABLES – a method of solving differential equations. Find the general solution of dy = 2x(y + 4) y > 0 ...
... SEPARATING THE VARIABLES – a method of solving differential equations. Find the general solution of dy = 2x(y + 4) y > 0 ...
24. Eigenvectors, spectral theorems
... be diagonalizable by having one (hence, at least two) of the zeros of its minimal polynomial lie in a proper field extension of k. For not finite-dimensional V , there are further ways that an endomorphism may fail to be diagonalizable. For example, on the space V of two-sided sequences a = (. . . , ...
... be diagonalizable by having one (hence, at least two) of the zeros of its minimal polynomial lie in a proper field extension of k. For not finite-dimensional V , there are further ways that an endomorphism may fail to be diagonalizable. For example, on the space V of two-sided sequences a = (. . . , ...
Stat 8112 Lecture Notes Weak Convergence in Metric Spaces
... Condition (ii) just repeats the definition of weak Rconvergence with uniformly continuous f replacing continuous f because f dPn is another notation for E{f (Xn )} when Pn is the law of Xn . The word “events” in (iii), (iv), and (v) refers to elements of the Borel sigma-field of the metric space. Of ...
... Condition (ii) just repeats the definition of weak Rconvergence with uniformly continuous f replacing continuous f because f dPn is another notation for E{f (Xn )} when Pn is the law of Xn . The word “events” in (iii), (iv), and (v) refers to elements of the Borel sigma-field of the metric space. Of ...
Lecture 11
... Two morphisms f and g are homotopic if there are maps k i : Ai −→ B i−1 such that f − g = dk + kd. If f and g are homotopic then hi (f ) = hi (g). A functor F from one abelian category U to another B is called additive if for any two objects A and B in U, the induced map Hom(A, B) −→ Hom(F A, F B), ...
... Two morphisms f and g are homotopic if there are maps k i : Ai −→ B i−1 such that f − g = dk + kd. If f and g are homotopic then hi (f ) = hi (g). A functor F from one abelian category U to another B is called additive if for any two objects A and B in U, the induced map Hom(A, B) −→ Hom(F A, F B), ...
(pdf)
... of continuity when considering functions from one matrix group to another. More importantly we can consider continuous homomorphisms, and from now on all homomorphisms will be assumed to be continuous. Then if we have two matrix groups G and H and a homomorphism φ : G → H then every curve,γ, in G wi ...
... of continuity when considering functions from one matrix group to another. More importantly we can consider continuous homomorphisms, and from now on all homomorphisms will be assumed to be continuous. Then if we have two matrix groups G and H and a homomorphism φ : G → H then every curve,γ, in G wi ...
(maximal) ideal in . Theorem
... any countable subfamily Fn n of the filter F . For a fine uniformity u f on ...
... any countable subfamily Fn n of the filter F . For a fine uniformity u f on ...