weak-* topology
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
... Let X be a locally convex topological vector space (over C or R), and let X ∗ be the set of continuous linear functionals on X (the continuous dual of X). If f ∈ X ∗ then let pf denote the seminorm pf (x) = |f (x)|, and let px (f ) denote the seminorm px (f ) = |f (x)|. Obviously any normed space is ...
MATH 730: PROBLEM SET 2 (1) (a) Let X be a locally compact
... (1) (a) Let X be a locally compact Hausdorff space. Then the intersection of an open subset of X and a closed subset of X is locally compact. (b) Let X be Hausdorff and Y ⊂ X be locally compact. Show that Y is the intersection of an open subset of X and a closed subset of X. (2) Prove or disprove: T ...
... (1) (a) Let X be a locally compact Hausdorff space. Then the intersection of an open subset of X and a closed subset of X is locally compact. (b) Let X be Hausdorff and Y ⊂ X be locally compact. Show that Y is the intersection of an open subset of X and a closed subset of X. (2) Prove or disprove: T ...
SECTION 2.3 If f : X → Y is a function, we say that X is the domain
... If f : X → Y is a function, we say that X is the domain, and Y is the co-domain. The range of f is defined as the set f (X) = {y ∈ Y | ∃x ∈ X, f (x) = y}. In general, if A ⊆ X is a subset of X, we define the image of A as the set f (A) = {y ∈ Y | ∃x ∈ A, f (x) = y}. The image of a set A is oftentime ...
... If f : X → Y is a function, we say that X is the domain, and Y is the co-domain. The range of f is defined as the set f (X) = {y ∈ Y | ∃x ∈ X, f (x) = y}. In general, if A ⊆ X is a subset of X, we define the image of A as the set f (A) = {y ∈ Y | ∃x ∈ A, f (x) = y}. The image of a set A is oftentime ...
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... X is not path-connected. Indeed, assume to the contrary that there exists a path γ : [0, 1] → X with γ(0) = ( π1 , 0) and γ(1) = (0, 0). Let c = inf {t ∈ [0, 1] | γ(t) ∈ X1 } . Then γ([0, c]) contains at most one point of X1 , while γ([0, c]) contains all of X1 . So γ([0, c]) is not closed, and ther ...
... X is not path-connected. Indeed, assume to the contrary that there exists a path γ : [0, 1] → X with γ(0) = ( π1 , 0) and γ(1) = (0, 0). Let c = inf {t ∈ [0, 1] | γ(t) ∈ X1 } . Then γ([0, c]) contains at most one point of X1 , while γ([0, c]) contains all of X1 . So γ([0, c]) is not closed, and ther ...
Lecture 1:
... 1. f(x) is a NOTATION It's defined to mean (in this case) x - 4 It does NOT mean multiplication 2. You'll know whether it means multiplication or as a stand-in for a function by the context (i.e., what's around it). In other words, if the problem says "Do something with the following expression, whe ...
... 1. f(x) is a NOTATION It's defined to mean (in this case) x - 4 It does NOT mean multiplication 2. You'll know whether it means multiplication or as a stand-in for a function by the context (i.e., what's around it). In other words, if the problem says "Do something with the following expression, whe ...
