GATE MA Functional Analysis PYQs & Answers — Page 2

Questions 11–20 of 47. Imported answers require independent review unless marked reviewed.

Consider the inner product space of all real-valued continuous functions defined on

with the inner product

. If

is orthogonal to all the polynomials having degree less than or equal to

, with respect to this inner product, then

is equal to _____ (in integer)

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

for

. Let

. For

, define

. Consider the following statements.

I. There exists a continuous linear functional

such that

II. There exists a continuous linear functional

such that

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

endowed with the inner product

Let

be given by

, where

Which of the following is/are correct?

Your answerSelect all correct options. You can choose more than one.

is a compact operator

The operator norm of

is 1

is a self-adjoint operator

is closed

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

be the normed space

, where

Let

and

be given by

for all

. Recall that a Hahn–Banach extension of

is a continuous linear functional

such that

and

, where

and

are the norms of

and

, respectively. Which of the following is/are true?

Your answerSelect all correct options. You can choose more than one.

is a Hahn–Banach extension of

admits infinitely many Hahn–Banach extensions to

admits exactly two distinct Hahn–Banach extensions to

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

denote the space of all real-valued Lebesgue square-integrable functions on

, with the usual norm

. Let

be the subspace of

consisting of all the polynomials of degree at most 1. Let

be such that

, and

. Then

(round off to TWO decimal places)

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

denote the set of all real valued continuous functions defined on

and

for all

. Let

Define

for all

. Denote

. Then the set

has

Your answerChoose one option.

NO element

exactly one element

exactly two elements

an infinite number of elements

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

and the norm

for

. Let

be defined by

denotes the norm of the linear functional

, then

is equal to _________.

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

. For a sequence

, define

. Let

and

be defined by

Then

Your answerChoose one option.

is a compact linear map and

is NOT a compact linear map

is NOT a compact linear map and

is a compact linear map

both

and

are compact linear maps

NEITHER

NOR

is a compact linear map

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

. For

, let

. Define

for all

. Then

Your answerChoose one option.

is continuous but

is NOT continuous

is NOT continuous but

is continuous

both

and

are continuous

NEITHER

NOR

is continuous

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

Let

. For a sequence

, define

Consider the subspace

. Let

denote the orthogonal complement of

in the Hilbert space

. Consider

If the orthogonal projection of

onto

is given by

for some

, then

equals __________.

You can attempt this question now. Sign in to view the answer key, explanations and discussion.Log in to check answer

GATE MA Functional Analysis PYQs

Question 61

Question 47

Question 48

Question 49

Question 57

Question 23

Question 32

Question 37

Question 38

Question 56