Basic questions Flashcards
1
Q
- What is a normed linear space? Provide examples of Banach and Hilbert spaces.
A
2
Q
- State and prove the projection theorem in Hilbert spaces.
A
3
Q
- Prove that a linear operator between two normed linear spaces is continuous iff it is bounded.
A
4
Q
- What is meant by the dual space of a Banach space?
A
5
Q
- State and prove the Riesz representation theorem in Hilbert spaces.
A
6
Q
- What is the content of the uniform boundedness principle?
A
7
Q
- Define the notion of weak convergence in Banach spaces.
A
8
Q
- Give the definition of the Sobolev space W^(k,p)(Ω).
A
9
Q
- How is the Fourier transform defined on S(R^n)
A
10
Q
- Provide the definition in terms of the Fourier transform of the Sobolev space H^k(R^n) and use this to generalize the definition to non-integer values of k.
A
11
Q
- Write out the Neumann series for the inverse of (I−T) when ||T|| < 1.
A
12
Q
- Define the spectrum of an operator in L(X), where X is a Banach space.
A
13
Q
- What is meant by the Hilbert space adjoint of a densely defined operator?
A
