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**Functional Analysis Solved MCQs**

Post Topic | Solved MCQs |

Book Code | MTH641 |

FinalTerm/MidTerm | Both |

University | Virtual University |

**Functional Analysis MCQs With Answers**

1: In an Inner Product space say X, if the sequences {n} and {y} are Cauchy, then (xn, Yn) is-

**necessarily a Cauchy Sequence in F**necessarily a Cauchy Sequence in X

not necessarily a Cauchy Sequence in F

not necessarily a Cauchy Sequence in X

2: Null space is also a

Linear functional.

Canonical mapping.**Vector space.**Metric space.

3: If T is a continuous linear operator from a normed space X to normed space Y, then Tis_

**bounded as well provided that X and Y are finite-dimensional**bounded as well irrespective of the dimensions of X and Y

unbounded as well Irrespective of dimensions of X and Y

unbounded as well provided that X and Y are Infinite dimensional

4: The mapping; T: V→ V defined by T(v) = a + v, where v EV is linear if

**a=0**a #0

5: If a sequence → in a normed space X, then for a bounded linear operator T

**Txn**TxnTx

6: Dual space is

71**R**11

R

7: For n-dimensional vector space and its dual space we have

R(X) = X = n

D(X) = X’ = n

dim(X) C dim(X)**dim(X) = dim(X) = n**

8: Since a bounded linear operator T from the normed space X to normed space Y is defined and given as; VED(T) > 0, such that ||T||||||, then ||T||

sup ||T||

ZED(T)**both sup ZED(T) sup RED(T) 70 and sup ||T|| ZED(T)**None of these

9: Dual space of a normed space is

**functional**.

Metric space.

Incomplete.

Banach space.

10: If T:c[0,1]-c[0,1], which is defined and given as; 7(x)={fo1k(t,T)x(t)dt: |x|<k,the such that ||Tx||sk||x|| →

**T is a non-linear bounded**T is linear unbounded

T is linearly bounded

T is non-linear unbounded

11: Since a bounded linear operator T from the normed space X to normed space Y is defined and given as; VxED(T) k> 0, such that ||T|| ||||, then ||T|| = 0⇒

**T = 0**€ (0)

B or D

k<0

12: The pair is called Banach space. (V.<…>)

Inner product space.

Metric space.**Complete space.**

13: Let T: XY be a linear operator, then the restriction of T is expressed as

To:Y B, BCY

To: B-BBCX

Tn:X-B, BCY**Tn: B-Y, BCX**

14: Let S and T be linear operators, we have

(ST) ST

(ST)-1=1**(ST) T-1S-1**(ST)-S-T-1

15: IFT: R2R is defined by T (x, y)-y-z, then T-1 (6)

((t,t-6):t}

{(6+t,t):t}

Both ((6+,t): ER} and {(t,t-6):t}**None of these**

16: The dot product is a

operator.

Mapping.**Functional.**Function.

17: If T and Ty are equal operators defined on a normed space X, then for any z X.Tiz necessarily

r=0.**1**

18: IfT: R2 R2 defined by T(x, y) = (y, -5x+4y), then T-1 (-2,7) =

(3,2)

(-3,2)

(3,-2)**T-1 (-2,7) = (3,2)**

**Conclusion**

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