Banach spaces - definitions and examples, continuous linear transformations, Banach theorem, natural embedding of X in X'' - One Line Questions
1.
What is the definition of the norm of a continuous linear functional f in X*? —
||f|| = sup { |f(x)| : ||x|| = 1 }
2.
What is the norm of the natural embedding map J: X -> X''? —
||J|| = 1
3.
The natural embedding J: X -> X'' maps an element x in X to: —
A linear functional on X'
4.
What is a continuous linear functional on a normed vector space X? —
A linear map f: X -> R (or C) such that |f(x)| <= M||x|| for some M.
5.
For a normed space X, the second dual X'' is always: —
A Banach space
6.
The natural embedding map J: X -> X'' preserves norms, meaning ||J(x)|| = ||x|| for all x in X. This makes J: —
An isometry
7.
The Open Mapping Theorem, closely related to the Banach theorem, states that a continuous linear surjective map between two Banach spaces is: —
An isomorphism
8.
The natural embedding J: X -> X'' maps x to the functional J(x) defined by J(x)(f) = f(x) for f in X*. This definition ensures that J is: —
A linear map
9.
What is a linear transformation T: X -> Y between normed spaces X and Y called if there exists a constant M such that ||T(x)|| <= M||x|| for all x in X? —
Bounded linear transformation
10.
Which of the following is a standard example of a reflexive Banach space? —
l_2
11.
The Closed Graph Theorem states that if T: X -> Y is a linear operator between Banach spaces X and Y, and its graph is closed, then T is: —
Continuous
12.
The concept of reflexivity in Banach spaces is related to the identification of the space with its: —
Second dual space via the natural embedding
13.
In a normed vector space X, what does completeness mean? —
Every Cauchy sequence converges to an element within X.
14.
If X is a Banach space and Y is a normed space, and {T_n} is a sequence of continuous linear operators from X to Y such that T_n(x) converges for each x in X, then T_n converges uniformly. —
False, pointwise convergence does not imply uniform convergence.
15.
The Banach space X is called separable if it contains a: —
Countable dense subset.
16.
Which of these spaces is NOT reflexive? —
l_1
17.
What is the definition of a Cauchy sequence in a normed vector space? —
For every epsilon > 0, there exists an N such that ||x_n - x_m|| < epsilon for all n, m > N.
18.
The Banach theorem is also known as: —
Uniform Boundedness Principle
19.
Which theorem is crucial for proving that the dual of a Banach space is a Banach space? —
Uniform Boundedness Principle
20.
Which of the following statements about the dual space X* is FALSE? —
If X is a reflexive Banach space, X* is reflexive.
21.
What is the fundamental property of a Banach space? —
It is a complete normed vector space.
22.
If X is a reflexive Banach space, what property does the natural embedding J: X -> X'' have? —
It is surjective.
23.
What is the natural embedding map J: X -> X''? —
J(x) = g_x, where g_x(f) = f(x) for all f in X'.
24.
Consider the space C([0,1]) with the supremum norm. This space is a Banach space. What is its dual space C([0,1])*? —
The space of finite signed measures on [0,1]
25.
Which property is NOT guaranteed for the natural embedding J: X -> X'' for any normed space X? —
Norm preservation (isometry)
26.
The Banach theorem (Uniform Boundedness Principle) is a powerful tool in functional analysis because it allows one to deduce uniform boundedness of a family of operators from: —
Pointwise boundedness and completeness of the domain space.
27.
The set of all continuous linear functionals on a normed vector space X forms a normed vector space itself, called the: —
Dual space
28.
Which of the following is a key application of the Uniform Boundedness Principle (Banach Theorem)? —
Establishing the completeness of the dual space.
29.
Consider a sequence of continuous linear operators {T_n} from a Banach space X to a normed space Y such that sup_n ||T_n(x)|| < infinity for every x in X. The Banach theorem states that: —
sup_n ||T_n|| < infinity
30.
Consider a sequence of linear operators T_n: X -> Y, where X is a Banach space. If T_n are continuous and sup_n ||T_n(x)|| < infinity for all x, what can be said about ||T_n||? —
sup_n ||T_n|| < infinity
31.
The natural embedding J: X -> X'' is always: —
Injective
32.
If T: X -> Y is a continuous linear operator between Banach spaces X and Y, and T is bounded, what can be said about the operator T:**: Y* -> X*? —
T** is also continuous and bounded.
33.
If T: X -> Y is a bounded linear operator between Banach spaces X and Y, and T is surjective, then T is an open map. This is a statement of: —
The Open Mapping Theorem
34.
Let T: X -> Y be a continuous linear operator between Banach spaces X and Y. The graph of T, G(T) = {(x, T(x)) : x in X}, is a closed subspace of X x Y. This is a consequence of: —
The Closed Graph Theorem
35.
Let X be a normed vector space. What is the space X''? —
The second dual space of X, consisting of continuous linear functionals on X'.
36.
Which condition is NOT equivalent to X being a reflexive Banach space? —
X is homeomorphic to its bidual X''.
37.
What does the Banach theorem imply about a pointwise bounded sequence of continuous linear operators from a Banach space X to a normed space Y? —
The sequence is uniformly bounded.
38.
If T: X -> Y is a bounded linear transformation, what is the norm of T, denoted ||T||? —
The supremum of ||T(x)|| over all x with ||x|| = 1.
39.
Which of the following is NOT a standard example of a Banach space? —
The space of all polynomials on [0, 1] P[0, 1]
40.
Let X be a Banach space. The dual space X* is also a Banach space under the operator norm. This is a fundamental result. —
True
41.
The natural embedding J: X -> X'' is always a linear transformation. —
True
42.
Let X be a Banach space and Y be a normed space. If T: X -> Y is a linear operator such that T(x_n) -> 0 whenever x_n -> 0 (i.e., T is continuous), then T is bounded. —
True, by the Closed Graph Theorem.
43.
The Banach theorem, also known as the Uniform Boundedness Principle, applies to a collection of operators between which types of spaces? —
A complete normed vector space and a normed vector space
44.
If X is a normed space, when is X reflexive? —
When the natural embedding J: X -> X'' is surjective
45.
If X is a normed space, which of the following is always true about the relationship between X, X*, and X**? —
X** is always complete.
46.
What condition is required for the natural embedding J: X -> X'' to be an isometry? —
No additional condition is needed; it is always an isometry.
47.
The space l_p for p=1 is a Banach space. Is it reflexive? —
No
48.
The space l_infinity is a Banach space. Is it reflexive? —
No
49.
Let X be a Banach space and Y be a normed space. If {T_n} is a sequence of continuous linear operators from X to Y such that sup_n ||T_n|| = infinity, does this imply that T_n(x) is unbounded for some x? —
Yes, by the Banach theorem (Uniform Boundedness Principle).