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  • Uniform Boundedness principle for bounded linear maps from Frechet . . .
    "Given a barrelled space X and a locally convex space Y, then any family of pointwise bounded continuous linear mappings from X to Y is equicontinuous (even uniformly equicontinuous) " Every Fréchet space is barreled: en wikipedia org wiki Barrelled_space $\endgroup$
  • Filter-dependent versions of the Uniform Boundedness Principle
    Given that the uniform boundedness principle comes as a consequence of the Baire category theorem, it is not surprising that the uniform F-boundedness principle will hold for every Fr echet space whenever Fis a (free) Baire lter (see Lemma 1 1 below) However, it was found in [3], that the notion of Baire lter is a rather restrictive one
  • A Proof of the Uniform Boundedness Principle - McGill University
    The Baire Category Theorem states that a complete metric space is non-meager in itself Consequently, we obtain the following easy corollary of the uniform bound-edness principle Corollary Let Xbe a Banach space and Ya normed space, over R or C Suppose that A L(X;Y) is non-empty collection and sup T2A kTxk< 1; 8x 2X: Then sup T2A kTk< 1:
  • Filter-dependent versions of the uniform boundedness principle
    The theory of uniform boundedness principles for (locally convex) topological vector spaces is traditionally phrased in terms of barrels and barrelled spaces To stay in line with this case where F is equal to the Fréchet filter C, we first introduce an F-analogue for the concept of barrelled space
  • Math 255A Lecture 11 Notes - GitHub Pages
    Theorem 1 1 (Banach-Steinhaus, uniform boundedness principle) Let F be a Frechet space, and let V be a locally convex space If L(F; V ) is such that for each x 2 F the set fT x : T 2 g V is bounded, then is equicontinuous On the other hand, if is not equicontinuous, then the set of all x 2 F such that fT x : T 2 g is bounded is a set of the
  • The Principle of Uniform Boundedness, and Friends
    fixed x ∈ X, B(x,y) is continuous in y) and linear, it is bounded For each fixed y ∈ Y, limn→∞ Tny = 0, since B(x,y) is continuous in x when y is held fixed Thus, for each fixed y ∈ Y, Tny n ∈ IN is bounded Hence, by the Principle of Uniform Boundedness, τ = sup kTnk n ∈ IN < ∞ so that, B(x n,yn) = kT
  • 3. 6. Uniform Boundedness Principle - East Tennessee State University
    Theorem The following application of the Uniform Boundedness Principle shows that B(X,Y) is closed under pointwise limits Theorem 3 11 Suppose that (Tn) is a pointwise convergent sequence of bounded linear operators from Banach space X to normed linear space Y That is, for each x ∈ X the sequence (Tnx) converges to an element Tx ∈ Y
  • Chapter 4 Uniform Boundedness and the Open Mapping Theorem - Springer
    Our first result is the principle of uniform boundedness or the Banach– Steinhaus theorem Theorem 4 1 (Banach–Steinhaus) Let Xbe a Banach space and let Y be a normed vector space Let {Tα | α∈ A} be a family of bounded linear operators from Xto Y Suppose that for each x∈ X, the set {Tαx| α∈ A} is a bounded subset of Y
  • Uniform Boundedness Principle -- from Wolfram MathWorld
    A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded " Symbolically, if sup||T_i(x)|| is finite for each x in the unit ball, then sup||T_i|| is finite





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