GATE MA concept:hilbert-spaces PYQs & Answers

Home
GATE
Mathematics
Topics
concept:hilbert-spaces

Attempt a question, record the concept and assumptions, then check the available answer. Rework mistakes after revising the relevant concepts. Topic tags can overlap; review the question context.

Questions 1–10 of 11. Imported answers require independent review unless marked reviewed.

Topic practice 11 questionsPage 1 of 2

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

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

Let

be an orthonormal basis for a Hilbert space

. Define

and

. Then

Your answerChoose one option.

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

Let

be an orthonormal basis of a complex Hilbert space

. Consider the following statements:

P: There exists a bounded linear functional

such that

for

Q: There exists a bounded linear functional

such that

for

Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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

Let

be a complex Hilbert space. Let

be such that

. Then

__________.

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

Let

be the Hilbert space of real valued square integrable functions on

equipped with the norm

. Consider the subspace

. For

, define

. Then

Your answerChoose one option.

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

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

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

Let

be the Hilbert space of all real valued square integrable functions on

with the usual inner product. Let

be the linear functional on

defined by

where

denotes the Lebesgue measure on

. Then

_________.

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

Let

be an orthonormal set in a Hilbert space

and let

be such that

. Consider the set

Then the maximum possible number of elements in

is _________.

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

Let

be an orthonormal basis of a Hilbert space

. Let

be given by

. For each

, define

. Then

Your answerChoose one option.

but for each

for each

but the sequence

is unbounded

there exist

such that

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

GATE MA concept:hilbert-spaces PYQs

How to practice concept:hilbert-spaces

Question 54

Question 56

Question 16

Question 9

Question 25

Question 33

Question 26

Question 47

Question 48

Question 39