SOLUTIONS TO PROBLEM SET 4 1. Without loss of generality
... generator of X, then the function g defined by g(x) = f (|x|) should belong to the infinitesimal generator of B. On the other hand, g is C20 (R) if and only if f is C20 (R+ ) with f 0 (0) = 0. Thus, the domain of the generator are twice continuously differentiable functions on [0, ∞) with zero deriv ...
... generator of X, then the function g defined by g(x) = f (|x|) should belong to the infinitesimal generator of B. On the other hand, g is C20 (R) if and only if f is C20 (R+ ) with f 0 (0) = 0. Thus, the domain of the generator are twice continuously differentiable functions on [0, ∞) with zero deriv ...
Document
... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...
... If the points P and Q have position vectors r(t) and r(t + h), then represents the vector r(t + h) – r(t), which can therefore be regarded as a secant vector. If h > 0, the scalar multiple (1/h)(r(t + h) – r(t)) has the same direction as r(t + h) – r(t). As h 0, it appears that this vector approac ...
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED
... Theorem 2.2. Let X be an F -space, {Yi }i∈I be a collection of vector spaces and pi be an αseminorm on Yi for a fixed α, (0 < α ≤ 1). Let {Ti }i∈I be a family of linear mappings Ti : X → Yi . Suppose {pi (Ti x) : i ∈ I} is bounded for each x ∈ X. Then there is an α-convex balanced open neighbourhood ...
... Theorem 2.2. Let X be an F -space, {Yi }i∈I be a collection of vector spaces and pi be an αseminorm on Yi for a fixed α, (0 < α ≤ 1). Let {Ti }i∈I be a family of linear mappings Ti : X → Yi . Suppose {pi (Ti x) : i ∈ I} is bounded for each x ∈ X. Then there is an α-convex balanced open neighbourhood ...
The weak dual topology
... Theorem 4.1 (Alaoglu). If X is a normed vector space, then the unit ball (X∗ )1 , in the topological dual space, is compact in the w∗ topology. Proof. The key ingredient in the proof will be the use of Proposition 4.4. Use the notations above. We know already that Θ : X∗ → Θ(X∗ ) is a homeomorphism ...
... Theorem 4.1 (Alaoglu). If X is a normed vector space, then the unit ball (X∗ )1 , in the topological dual space, is compact in the w∗ topology. Proof. The key ingredient in the proof will be the use of Proposition 4.4. Use the notations above. We know already that Θ : X∗ → Θ(X∗ ) is a homeomorphism ...
Solutions to some problems (Lectures 15-20)
... ~ around the unit circle in the xy-plane, (b) Find the line integrals of F~ , G, centered at the origin, and traversed counterclockwise. (c) For which of the three vector fields can Green’s Theorem be used to calculate the line integral in part (b)? Why? ...
... ~ around the unit circle in the xy-plane, (b) Find the line integrals of F~ , G, centered at the origin, and traversed counterclockwise. (c) For which of the three vector fields can Green’s Theorem be used to calculate the line integral in part (b)? Why? ...
Weak topologies - SISSA People Personal Home Pages
... is separating, then the topology generated by P makes X into a locally convex topological vector space. We denote the topology generated by P by σ(X, X ∗ ). Remark 1.1. If X is locally convex, for example a Fréchet space, normed or Banach space, then P is separating by the Hahn-Banach theorem. Lemm ...
... is separating, then the topology generated by P makes X into a locally convex topological vector space. We denote the topology generated by P by σ(X, X ∗ ). Remark 1.1. If X is locally convex, for example a Fréchet space, normed or Banach space, then P is separating by the Hahn-Banach theorem. Lemm ...
Nuclear Space Facts, Strange and Plain
... simply because the topology τU is contained in the full topology τ of X. For the converse, assume that the topology τ of X has a local base N consisting of convex, balanced neighborhoods U of 0 for which ρU is a norm, i.e. that every neighborhood of 0 contains as subset some neighborhood in N . Supp ...
... simply because the topology τU is contained in the full topology τ of X. For the converse, assume that the topology τ of X has a local base N consisting of convex, balanced neighborhoods U of 0 for which ρU is a norm, i.e. that every neighborhood of 0 contains as subset some neighborhood in N . Supp ...
Continuity of the coordinate map An “obvious” fact
... The collection of open sets in Rm is a Hausdorff topology. A property of X that can be expressed in terms of its topology is called a topological property. 9 Definition Let X and Y be topological spaces and let f : X → Y . Then f is continuous if the inverse image of open sets are open. That is, if ...
... The collection of open sets in Rm is a Hausdorff topology. A property of X that can be expressed in terms of its topology is called a topological property. 9 Definition Let X and Y be topological spaces and let f : X → Y . Then f is continuous if the inverse image of open sets are open. That is, if ...
4 Vector Spaces
... 4. Let S be any set and F (S, K) denote the set of all functions from S to K. Given any two functions f1 , f2 : S −→ K, and a scalar α ∈ K we define f1 + f2 and αf1 by (f1 + f2 )(x) = f1 (x) + f2 (x); (αf1 )(x) = αf1 (x), x ∈ S. Then it is easily verified that F (S, K) becomes a vector space. The ze ...
... 4. Let S be any set and F (S, K) denote the set of all functions from S to K. Given any two functions f1 , f2 : S −→ K, and a scalar α ∈ K we define f1 + f2 and αf1 by (f1 + f2 )(x) = f1 (x) + f2 (x); (αf1 )(x) = αf1 (x), x ∈ S. Then it is easily verified that F (S, K) becomes a vector space. The ze ...
Definition 1. Cc(Rd) := {f : R d → C, continuous and with compact
... (5) Fourier transform F, F −1 , to be discussed only later: our normalization will be that given for f ∈ Cc (Rd ) by the integral Z ...
... (5) Fourier transform F, F −1 , to be discussed only later: our normalization will be that given for f ∈ Cc (Rd ) by the integral Z ...
Document
... numbers. Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. • The definition of a vector space specifies neither the nature of the vectors nor the operations. Any kind of object can be a ...
... numbers. Vector spaces in which the scalars are complex numbers are called complex vector spaces, and those in which the scalars must be real are called real vector spaces. • The definition of a vector space specifies neither the nature of the vectors nor the operations. Any kind of object can be a ...
Supplement: Basis, Dimension and Rank
... need to examine whether S is linearly independent or S spans R 3 . We don’t need to examine S being both linearly independent and spans V. Example: ...
... need to examine whether S is linearly independent or S spans R 3 . We don’t need to examine S being both linearly independent and spans V. Example: ...
NOTES ON DUAL SPACES In these notes we introduce the notion
... v ∈ V is an element such that f (v) = 0 for all f ∈ V ∗ , then v = 0. To see this, let (v1 , . . . , vn ) be a basis of V and let (f1 , . . . , fn ) be the dual basis. Write v as v = a1 v1 + · · · + an vn . By assumption, we have that fi (v) = 0 for all i. But by the definition of fi , fi (v) = ai . ...
... v ∈ V is an element such that f (v) = 0 for all f ∈ V ∗ , then v = 0. To see this, let (v1 , . . . , vn ) be a basis of V and let (f1 , . . . , fn ) be the dual basis. Write v as v = a1 v1 + · · · + an vn . By assumption, we have that fi (v) = 0 for all i. But by the definition of fi , fi (v) = ai . ...
Ultraproducts of Banach Spaces
... Real-valued ultralimits exend our normal definition of limit and have (almost) all the standard properties of normal limits we know and love: they are linear, multiplicative, and respect order. They are not shift invariant: limU xn 6= limU xn+1 in general (since either the even numbers or the odd nu ...
... Real-valued ultralimits exend our normal definition of limit and have (almost) all the standard properties of normal limits we know and love: they are linear, multiplicative, and respect order. They are not shift invariant: limU xn 6= limU xn+1 in general (since either the even numbers or the odd nu ...
Talk 2
... Let V be a base of circled neighborhoods in F . Then ∀V ∈ V : ∃W ∈ V|W + W ⊂ V Let be H1 , H2 ∈ K, then H1−1 (W ) and H2−1 (W ) are neighborhoods of zero in E. H1−1 (W ) ∩ H2−1 (W ) ⊂ (H1 + H2 )−1 (V ), so (H1 + H2 )−1 (V ) is an open neighborhood of 0 in E. Since V is an arbitrary element of a base ...
... Let V be a base of circled neighborhoods in F . Then ∀V ∈ V : ∃W ∈ V|W + W ⊂ V Let be H1 , H2 ∈ K, then H1−1 (W ) and H2−1 (W ) are neighborhoods of zero in E. H1−1 (W ) ∩ H2−1 (W ) ⊂ (H1 + H2 )−1 (V ), so (H1 + H2 )−1 (V ) is an open neighborhood of 0 in E. Since V is an arbitrary element of a base ...
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910).Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.Lebesgue spaces have applications in physics, statistics, finance, engineering, and other disciplines.