GATE MA 2024 Paper 1 — page 17

Questions 49–51 of 65. Imported answers require review unless explicitly marked reviewed.

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

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Let

be a basis for a topology

. Which of the following is/are correct?

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

Every

with

is an open set in

Every

with

is compact in

is a first-countable space

is a second-countable space

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Let

be two non-zero, non-identity

-linear transformations. Assume

. Which of the following is/are TRUE?

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

is necessarily invertible

and

are similar if

and

are similar if

has only 0 and 1 as eigenvalues

is necessarily diagonalizable

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2024 · Paper 1

Question 49

Question 50

Question 51