Functional Analysis Answers

Let x1 x2 … . . xn, be an orthonormal set in X and 𝑘1 ,𝑘2,. … … … 

𝑘nbe scalars

having absolute value 1. Then 𝑘1x1 + 𝑘 2x 2+ ⋯ … 𝑘 nxn= x1 + ⋯ + 

xn

Show that an idem potent operator on a Hilbert space H is a

projection on H

iff it is normal.

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]