Download PROBLEM SET ON ANALYSIS 1. Prove or disprove the following

PROBLEM SET ON ANALYSIS
1. Prove or disprove the following statements:
(1) the set of continuous functions on [0, 1] is dense in L∞ [0, 1].
(2) L∞ [0, 1] is a separable metric space.
2. Let X and Y be Banach spaces, and L(X, Y ) the space of bounded linear maps
from X to Y .
(1) Show that L(X, Y ) is a Banach space.
(2) Show that the set of invertible operators is open in L(X, Y ).
3. Let A : X → Y be a continuous linear map from one Banach space to another.
Assume that there is a positive number k such that ||Ax|| ≥ k||x|| for all x ∈ X.
Prove that A(X) is closed in Y .
4. Let T : l2 (Z) → l2 (Z) be the shift operator: T (x)n = xn+1 for any x = (xn ) ∈
l2 (Z). Show that for all λ ∈ C, the operator T − λE is injective and has dense range.
5. Prove that l∞ is not a separable Banach space.
6. Let H be a Hilbert space and A a bounded linear transformation of H. Let
P = E + A∗ A.
(1) Show that ||u|| ≤ ||P u||.
(2) Show that P has a bounded inverse.
7. Show that C ∞ [−1, 1] is not dense in Lip[−1, 1], where Lip is the Banach space
of functions bounded with respect to the Lipschitz norm
|f (x) − f (y)|
|f |Lip = sup |f | + sup
.
|x − y|
x6=y
8. Let H1 be the Sobolev space of all functions f ∈ L2 [0, 1] such that
∞
X
||f ||21 =
(1 + n2 )|fˆ(n)|2 < ∞,
n=−∞
where fˆ(n) are the Fourier coefficients of f . Show that there is a constant C > 0
such that ||f ||∞ ≤ C||f ||1 .
9. Let H be a separable Hilbert space. Prove that any bounded sequence in H has
a weakly convergent subsequence.
10. Given any non-negative sequence y = (y1 , y2 , . . . , yn , . . . ), set Xy = {x ∈
l2 | |xn | ≤ yn ∀n ≥ 1}. Prove that Xy is compact iff y ∈ l2 .
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11. Prove or disprove: if T : H1 → H2 is a bounded linear operator between Hilbert
spaces such that T (H1 ) is dense in H2 , then T is onto.
P
12. Suppose that (an ) is a sequence with the property that
|an bn | < ∞ whenever
(bn ) ∈ l2 . Show that (an ) ∈ l2 . Hint: consider the operator T : l2 → l1 given by
T (b1 , b2 , . . . ) = (a1 b1 , a2 b2 , . . . ).
13. Let X, Y and Z be Banach spaces, and let j : Y → Z be an injective bounded
linear operator. Suppose that f : X → Y is a linear operator such that j ◦ f is
bounded. Prove that f is bounded.
14. Let {ϕn } be an orthonormal basis in L2 (X), where X has finite measure. Prove
that {(x, y) 7→ ϕn (x)ϕm (y)} is an orthonormal basis in L2 (X × X).
15. Consider a map T : X → X from a Banach space X to itself such that ||T (x) −
T (y)|| ≤ ||x − y|| for all x, y ∈ X and T (M ) ⊂ M for some non-empty bounded
closed convex set M ⊂ X. Show that, for every ε > 0, there exists a point x ∈ M
such that ||T x − x|| < ε.
16. Let 1 < p, q < ∞. Show that Lp (R+ ) 6⊆ Lq (R+ ).
17. Let A be a bounded linear operator on a Hilbert space H. Show that
(1) ||A|| = ||A∗ ||,
(2) AA∗ − A∗ A cannot be the identity operator.
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