functional analysis ii

functional analysis ii

University

8 Qs

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functional analysis ii

functional analysis ii

Assessment

Quiz

Mathematics

University

Practice Problem

Hard

Created by

MANJULA D

Used 2+ times

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8 questions

Show all answers

1.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Given a metric space (X, d), which of the following is true?

d(x, y) = 0 for all x, y in X.
d(x, y) > 0 for all x ≠ y in X.
d(x, y) ≥ 0 for all x, y in X.
d(x, y) = 0 if and only if x = y.

2.

MULTIPLE CHOICE QUESTION

1 min • 1 pt

Create: How can we extend the Banach space concept to other spaces in functional analysis?

By defining norms on infinite-dimensional spaces and ensuring their completeness
By defining norms on finite-dimensional spaces and ensuring their completeness
By defining metrics on infinite-dimensional spaces and ensuring their incompleteness
By defining metrics on finite-dimensional spaces and ensuring their incompleteness

3.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the difference between a metric space and a normed space?

A metric space has a notion of distance, while a normed space has a notion of magnitude.
A normed space has a notion of distance, while a metric space has a notion of magnitude.
A metric space has both a notion of distance and magnitude, while a normed space has only a notion of magnitude.
A normed space has both a notion of distance and magnitude, while a metric space has only a notion of distance.

4.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

What is the significance of the triangle inequality in a metric space?

It ensures that the distance between two points is always zero.
It ensures that the distance between two points is always non-negative.
It ensures that the distance between two points satisfies the triangle inequality.
It ensures that the distance between two points satisfies the triangle equality.

5.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

How does the notion of a metric affect the topology of a space?

It does not affect the topology of a space.
It affects the topology of a space by defining the notion of an open set.
It affects the topology of a space by defining the notion of a closed set.
It affects the topology of a space by defining both the notions of open and closed sets.

6.

MULTIPLE CHOICE QUESTION

20 sec • 1 pt

What is a metric space?

A set with a binary operation
A set with a notion of distance
A set with a binary operation and a notion of distance
None of the above

7.

MULTIPLE CHOICE QUESTION

20 sec • 1 pt

Which of the following is not a property of a metric?

Non-negativity
Symmetry
Triangle inequality
Commutativity

8.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

In a metric space, what is the triangle inequality?

The sum of the lengths of any two sides of a triangle is greater than or equal to the length of the third side.
The product of the lengths of any two sides of a triangle is greater than or equal to the length of the third side.
The sum of the lengths of any two sides of a triangle is less than or equal to the length of the third side.
The product of the lengths of any two sides of a triangle is less than or equal to the length of the third side.