Functional Analysis Answers

Say True or False with proof

Q1. If 𝑓(𝑥) = 2|𝑥 − 1| and 𝑔(𝑥) = 3𝑥 − 10,

then 𝑓𝑜𝑔(1) = 12.

Q2. (√2, 1,1/2)∈ 𝑸 × 𝒁 × 𝑹.

Q3. The domain of the function f defined by 𝑓(𝑥)

=√{(3-x)/(x-2)} is R-{2}.

Represent the rational function f(x)=1/3x-1 using its table values and graph

For an example of a discontinuous linear functional on a normed linear space

prove ||Ax|| < ||A|| ||x||

Example of metric space but not norm

Let (X,d) be a metric space and consider the subset A ⊂ X and B ⊂ X . Show that the closure satisfies the following property closure( A ∪ B) =closure(A) ∪ closure(B)

Show that a norm on a vector space X is a sublinear functional on X

If a function f is not defined at x = a then the limit lim f(x) as x approaches a never exists. true or false

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.