GATE MA topic:functional-analysis PYQs & Answers — Page 4

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topic:functional-analysis

Questions 31–40 of 47. Imported answers require independent review unless marked reviewed.

Topic practice 47 questionsPage 4 of 5

Let

denote the space of all real-valued continuous functions on

equipped with the supremum norm

. Let

be the linear operator defined by

Then

Your answerChoose one option.

is not invertible

is surjective

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Let

denote the space of all real-valued continuous functions on

equipped with the supremum norm

. Let

be such that

For

, let

. If

, then

Your answerChoose one option.

the closure of

is compact

is closed and bounded

is bounded but not totally bounded

is compact

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Let

be equipped with the norm

and let

be the linear functional on

given by

Then

is equal to __________

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Let

be an orthonormal basis for a separable Hilbert space

with the inner product

. Define

for

. Then

Your answerChoose one option.

the closure of the span

equals

for all

is an orthogonal subset of

there does not exist nonzero

such that

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For

, let

and

be the bounded linear operators defined by

, where

and

. Then

Your answerChoose one option.

does not converge to

converges to zero as

for all

converges to zero as

for each non-zero

, there exists a continuous linear functional

such that

does not converge to

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Let

and let

. For

, define

. Then

Your answerChoose one option.

defines a norm on

for all

converges to

pointwise on

, then

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Let

and

be normed linear spaces, and let

be any bijective linear map with closed graph. Then which one of the following statements is TRUE?

Your answerChoose one option.

The graph of

is equal to $X\times Y

is continuous

The graph of

is closed

is continuous

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Let

be a normed linear space with the norm

Let

be the bounded linear functional defined by

Then

is equal to _____ (round off to 3 places of decimal).

(

is the set of all complex numbers.)

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Let

be an infinite matrix over

(the set of all complex numbers) such that

(i) for each

(the set of all natural numbers), the

row

is in

and

(ii) for every

is summable for all

, and

, where

Let the set of all rows of

be denoted by

. Consider the following statements:

is a bounded set in

is a dense set in

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both P and Q

P only

Q only

Neither P nor Q

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Consider the subspace

of the normed linear space

. If

is a bounded linear functional on

, defined by

, then which one of the following sets is equal to

(

is the set of all complex numbers,

and

)

Your answerChoose one option.

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GATE MA topic:functional-analysis PYQs

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