PROBLEM SET 9
Problem 1.
Let T : (C[0, 1], k·k∞ ) → (C[0, 1], k·k∞ ) be a linear operator, that carries pointwise
convergent sequences of functions of C[0, 1] to pointwise convergent sequences of
functions of C[0, 1], that is for every sequence (fn ) of C[0, 1] that converges pointwise to some f in C[0, 1], the sequence (T fn ) of C[0, 1] converges pointwise to T f .
Prove that T is a bounded operator.
Problem 2.
Let (X, k · k) be a normed space and x0 ∈ X, x0 6= 0X . We define the mapping
Tx0 : X ∗ → R : x∗ 7→ x∗ (x0 ) .
(i) Prove that Tx0 is linear, bounded and calculate its norm.
(ii) Prove that Tx0 is onto R.
Problem 3.
Let (X, k · k) be a normed separable space and (xn ) be a dense sequence in the
sphere SX . We define the operator
∗
x (xn )
.
T : X ∗ → c0 : T (x∗ ) =
n
Prove that
(i) T is well-defined, linear and 1-1
(ii) T is bounded and calculate its norm.
Problem 4.
Let (X, k · kX ) be a normed space and x1 , . . . , xn , x ∈ X. Prove that x is not
a linear combination of x1 , . . . , xn if and only if there exists x∗ ∈ X ∗ such that
x∗ (x1 ) = · · · = x∗ (xn ) = 0
and x∗ (x) 6= 0.
Problem 5.
Let (X, k · kX ), (Y, k · kY ) be normed spaces and T : X → Y a linear bounded
operator. We define the adjoint operator T ∗ of T , as
T ∗ : Y ∗ → X ∗ : T ∗ (y ∗ ) := y ∗ ◦ T .
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Prove that
(i) T ∗ is well-defined, linear and bounded.
(ii) kT ∗ k = kT k .
Problem 6.
Let (X, k · kX ), (Y, k · kY ) be normed spaces and T : X → Y a linear bounded
operator. We define the adjoint operator T ∗ of T , as
T ∗ : Y ∗ → X ∗ : T ∗ (y ∗ ) := y ∗ ◦ T .
Prove that
(i) T ∗ is 1-1 if and only if T has a dense range of values in Y (Hint: use the result
of Problem 3 of the Problem-Set 8).
(ii) if T : X → Y is an isomorphic embedding, then T ∗ is onto X ∗ .
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