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Let g ∈ Y ∗ and define T ∗ : X ∗ → F by the formula (T ∗ g)(x) = g(T x) for x ∈ X. Then, T ∗ g ∈ X ∗ . For T ∗ g = g ◦ T is a composition of continuous linear maps, 54 2. NORMED LINEAR SPACES AND BANACH SPACES T g X ✲Y ✲ F ✒ T ∗g and so is itself continuous and linear. Moreover, if g ∈ Y ∗ , x ∈ X, then |T ∗ g(x)| = |g(T x)| ≤ g ≤ g = T Y∗ g Y∗ Y∗ B(X,Y ) T Tx x Y X x B(X,Y ) X . Hence, not only is T ∗ g bounded, but T ∗g X∗ ≤ T B(X,Y ) g Y ∗ . Y ∗ → X ∗ . 23) T∗ Thus we have defined a correspondence : is itself a bounded linear map, which is to say T ∗ ∈ B(Y ∗ , X ∗ ).

If (X, d) and (Y, ρ) are two metric spaces and f : X → Y is a function, then f is continuous if for any x ∈ X and ε > 0, there exists a δ = δ(x, ε) > 0 such that d(x, y) ≤ δ implies ρ(f (x), f (y)) ≤ ε . If (X, · X ) and (Y, · Y ) are NLS’s, then they are simultaneously linear spaces and metric spaces. 3) to be an interesting class of mappings that are consistent with both the algebraic and metric structures of the underlying spaces. Continuous linear mappings between NLS’s are often called bounded operators or bounded linear operators or continuous linear operators.

In particular, D has all its limit points, so D is closed. 6. UNIFORM BOUNDEDNESS PRINCIPLE 49 Example. Closed does not imply bounded in general, even for linear operators. Take X = C(0, 1) with the max norm. Let T f = f for f ∈ D = C 1 (0, 1). Consider T as a mapping of D into X. T is not bounded . Let fn (x) = xn . Then fn = 1 for all n, but T fn = nxn−1 so T fn = n. X → f and fn → g. Then, by the Fundamental T is closed . Let {fn }∞ n=1 ⊂ D and suppose fn − Theorem of Calculus, t fn (τ ) dτ fn (t) = fn (0) + 0 for n = 1, 2, .

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Lectures on algebraic solutions of hypergeometric differential equations by Michihiko Matsuda

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