Functional Analysis Answers

Prove that a Hilbert space is seperableiff every ortho normal set in

H is countable.

Show that the self adjoint operator is continuous map

) Let T be a normal operator on a finite dimensional Hilbert space H

with spectrum𝜆1, 𝜆2, … … , 𝜆𝑚 , . Then prove that

i) T is self - adjoint⟺ each 𝜆𝑖,,is real

ii) T is positive ⟺ each 𝜆𝑖 ≥ 0

iii) T is unitary ⟺ 𝜆𝑖 =1 for each .

Prove that 𝑇1 and 𝑇2 are self adjoint operators on a Hilbert space

H, prove that 𝑇1 𝑇2 +𝑇2 𝑇1 is self adjoint

1.Show that an operator T on a Hilbert Space H is unitary iff

T(𝑒𝑖) complete orthonormal set whenever 𝑒𝑖

is.

Let X be a normed linear space and Y a closed subspace of X 

with Y ≠ X ,if 0 < 𝑟 < 1 prove that there exist 

𝑋r element of X such that ||X|| = 1And 𝑟 < 𝑑 𝑋𝑟

, 𝑌 ≤ 1

Find the norm of the linear functional f defined by

𝑓 𝑥 =integral -1 to 0𝑥 𝑡 𝑑𝑡 − integral 0 to 1𝑥 𝑡 𝑑𝑡

where𝑥 ∈ [−1 , 1]

Prove that

i) 𝑑 𝑎𝑥 , 𝑎𝑦 = 𝑎 d(x,y)

ii)𝑑 𝑎 + 𝑥 , 𝑎 + 𝑦 = 𝑑 𝑥 , 𝑦

where d is a metric induced by on a normed space X

Proof whether the following operations are inner product operations:

⟨x, y⟩ = 2x₁y₁ − x₁y₂ − x₂y₁ + 2x₂y₂, x=(x₁, x₂), y=(y₁, y₂)

Show that the space L[a,b] of all square integrable functions on the interval [a,b] is a linear space over a vector field R