Hilbert spaces - orthonormal bases, conjugate space, adjoint of an operator, projections - One Line Questions
1.
The adjoint operator T* of a bounded linear operator T: H1 -> H2 is defined by the relation: —
<T(x), y> = <x, T*(y)> for all x in H1, y in H2
2.
Consider an orthonormal basis {e_n} in a separable Hilbert space H. For any two vectors x and y, Parseval's identity for the inner product is: —
<x, y> = Sum(<x, e_n>) * <y, e_n>
3.
If P is a projection onto a closed subspace M of a Hilbert space H, then for any x in H, P(x) is the vector in M that minimizes the distance to x. This means: —
||x - P(x)|| <= ||x - m|| for all m in M
4.
What is the adjoint of the zero operator 0 on a Hilbert space H? —
0
5.
What is the adjoint of the identity operator I on a Hilbert space H? —
I
6.
If T is a bounded linear operator on a Hilbert space H, then the adjoint T* satisfies ||T*|| = ||T||. This implies that the algebra of bounded linear operators is: —
A C*-algebra.
7.
The Gram-Schmidt process is used to construct: —
An orthonormal basis from any linearly independent set.
8.
For a Hilbert space H, the set of all bounded linear operators B(H) equipped with the operator norm and the adjoint operation forms: —
All of the above.
9.
If T is a bounded linear operator on a Hilbert space H, then the operator T*T is: —
Always self-adjoint and positive semi-definite.
10.
If T is a bounded linear operator on a Hilbert space H, then the operator TT* is: —
Always self-adjoint and positive semi-definite.
11.
Let P1 and P2 be two projection operators on a Hilbert space H. When is their sum P1 + P2 also a projection? —
If P1 * P2 = 0 (i.e., their ranges are orthogonal).
12.
Let P1 and P2 be two projection operators on a Hilbert space H. When is their product P1 * P2 also a projection? —
If P1 and P2 commute (P1 * P2 = P2 * P1).
13.
If T is a linear operator on a Hilbert space H such that <Tx, y> = <x, Ty> for all x, y in H, then T must be: —
Self-adjoint.
14.
A projection operator P in a Hilbert space projects onto a subspace. What kind of subspace is it? —
A closed subspace.
15.
Let {e_n} be an orthonormal basis for a separable Hilbert space H. For any vector x in H, the Fourier coefficients are given by: —
c_n = <x, e_n>
16.
The concept of an orthonormal basis is crucial for: —
All of the above.
17.
A projection P is called an orthogonal projection if its range and kernel are: —
Orthogonal complements of each other.
18.
The Riesz Representation Theorem establishes a canonical isomorphism between a Hilbert space H and its conjugate space H*. What does this theorem state? —
Every bounded linear functional f in H* corresponds to a unique vector y in H such that f(x) = <x, y> for all x in H.
19.
A Hilbert space is separable if it contains a: —
Countable dense subset.
20.
For a bounded linear operator T from a Hilbert space H1 to a Hilbert space H2, what is the conjugate space of H1? —
The space of bounded linear functionals on H1 (H1*).
21.
Let T be a bounded linear operator from Hilbert space H1 to Hilbert space H2. The adjoint operator, denoted by T*, maps from which space to which space? —
H2 to H1
22.
The conjugate space of the space of continuous functions C([a, b]) with the sup norm is NOT necessarily a Hilbert space. This highlights the importance of the: —
Inner product structure.
23.
The conjugate space of a finite-dimensional Hilbert space is: —
Isomorphic to the space itself.
24.
What is the significance of an orthonormal basis in a Hilbert space? —
It allows for any vector to be represented as an infinite series of projections onto the basis vectors.
25.
What is the defining characteristic of a Hilbert space? —
It is a complete inner product space.
26.
What is the conjugate space of the space L^2([a, b])? —
L^2([a, b]) itself.
27.
What is the kernel (null space) of a projection operator P onto a closed subspace M? —
The orthogonal complement of M (M^perp).
28.
The conjugate space H* of a Hilbert space H is itself a: —
All of the above.
29.
Consider the Hilbert space L^2([0, 1]). The operator of multiplication by x, (Mf)(t) = t*f(t), is: —
A bounded self-adjoint operator.
30.
A projection P onto a closed subspace M is orthogonal if and only if: —
The range of P is M and the kernel of P is M^perp.
31.
What property does a projection operator P on a Hilbert space satisfy? —
P* = P and P^2 = P
32.
Which of the following is NOT a Hilbert space? —
C([0, 1]) with the sup norm.
33.
What is the spectral theorem for self-adjoint operators on a Hilbert space about? —
All of the above.
34.
Which theorem guarantees that every closed subspace of a Hilbert space is the range of some orthogonal projection? —
Projection Theorem
35.
In a Hilbert space, if {e_i} is an orthonormal set, then for any vector x, the projection of x onto the subspace spanned by {e_i} is given by: —
Sum(<x, e_i>) * e_i
36.
Bessel's inequality for an orthonormal set {e_i} in a Hilbert space H and any vector x in H states: —
Sum(|<x, e_i>|^2) <= ||x||^2
37.
If T is a bounded linear operator on a Hilbert space H, what is the definition of the operator norm ||T||? —
sup { ||Tx|| : ||x|| = 1 }
38.
If T is a bounded linear operator on a Hilbert space H, and T is normal (i.e., T*T = TT*), then: —
T is diagonalizable by a unitary operator.
39.
An operator T on a Hilbert space H is called self-adjoint if: —
T* = T
40.
What is the adjoint of an operator T defined by <Tx, y> = x_1*y_2 + x_2*y_1 on R^2? —
T* is the same as T.
41.
If H is a Hilbert space and T is a bounded linear operator on H, then T is invertible if and only if: —
All of the above.
42.
Consider the Hilbert space l^2 of square-summable sequences. Let T be an operator such that T((x_n)) = (0, x_1, x_2, ...). What is T*? —
T*((y_n)) = (y_2, y_3, ...)
43.
If T is a bounded linear operator on a Hilbert space H, what is the relationship between T** and T? —
T** = T
44.
If T is a bounded linear operator on a Hilbert space H, and T is unitary, then: —
T*T = I and TT* = I
45.
In a Hilbert space, what property must an orthonormal set satisfy? —
The inner product of any two distinct vectors is 0, and the norm of each vector is 1.
46.
The set of all bounded linear operators on a Hilbert space H forms an algebra, which is a Banach algebra. What is the norm in this algebra? —
The operator norm.
47.
What is the conjugate space (dual space) of a Hilbert space H, denoted by H*? —
The space of all bounded linear functionals on H.
48.
What is the conjugate space of a finite-dimensional vector space V? —
The dual space V*.
49.
When does Bessel's inequality become Parseval's identity for a complete orthonormal set {e_i} in a Hilbert space H and a vector x in H? —
When the set is countable and the sum converges to ||x||^2.