GATE MA topic:functional-analysis PYQs & Answers

Questions 1–10 of 47. Imported answers require independent review unless marked reviewed.

Let

be the space of all continuously differentiable real valued functions on

. Define the following norms on

and

. Which one of the following is TRUE?

Your answerChoose one option.

is a Banach space.

is not a Banach space.

is a Banach space.

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Let

where

and

be a linear transformation defined as

. Then the operator norm of

is equal to ________. (answer in integer)

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Let

with the supremum norm. Let

be given by

. Which one of the following is TRUE?

Your answerChoose one option.

is bounded but not one-to-one.

is one-to-one but not bounded.

is bounded and the inverse (from the range of

) exists but not bounded.

is bounded and the inverse (from the range of

) is bounded.

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Let

and

be subsets of a normed linear space

, and

. Which one of the following is NOT TRUE?

Your answerChoose one option.

If either of

is open, then

is open.

and

are convex, then

is convex.

and

are closed, then

is closed.

and

are connected, then

is connected.

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Let

be any normed linear space and

be the dual space of

. Which of the following statements is/are TRUE?

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

is separable and

is non-separable, then

must be reflexive.

is separable and

is non-separable, then

cannot be reflexive.

is reflexive, then

is reflexive.

is separable, then

is separable.

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Let

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

. Let

be defined as follows:

Then the value of

is equal to ________. (answer in integer)

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Let

with the norm

, where

is the space of all real-valued continuous functions on

. Let

with the norm

. Let

and

be the closed unit balls in

and

centred at the origin, respectively. Consider

and

given by

Consider the following statements:

S1:

is attained at a point of

S2:

is attained at a point of

Then, which one of the following is correct?

Your answerChoose one option.

S1 is TRUE and S2 is FALSE

S2 is TRUE and S1 is FALSE

both S1 and S2 are TRUE

neither S1 nor S2 is TRUE

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Consider the following two spaces:

, the space of all real-valued continuous functions defined on

equipped with the norm

, the space of all real-valued continuous functions defined on

equipped with the norm

Let

be the linear span over

of all the Legendre polynomials. Then, which one of the following is correct?

Your answerChoose one option.

is dense in

but not in

is dense in

but not in

is dense in both

and

is dense neither in

nor in

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Let

. Define

It is given that

is a Banach space with respect to the norm

Let

be the space of all real-valued continuous functions on

with the norm

. If

is the map

, then which of the following is/are TRUE?

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

is a compact linear map

Image of

is closed in

Image of

is dense in

is not a bounded linear map

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Let

be an orthonormal set of vectors in a real Hilbert space

with inner product

. Let

, and let

be the linear span of

over

. For

, let

. Then, which of the following is/are TRUE?

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

is the orthogonal projection of

onto

is the orthogonal projection of

onto

is orthogonal to

for all

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GATE MA topic:functional-analysis PYQs

How to practice topic:functional-analysis

Question 19

Question 32

Question 42

Question 45

Question 53

Question 60

Question 17

Question 45

Question 50

Question 54