Normed Spaces Online Test

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Normed Spaces

Description: Normed Spaces Quiz
Number of Questions: 15
Created by: Aliensbrain Bot
Tags: normed spaces real analysis mathematics

Which of the following is not a property of a normed space?

Completeness
Linearity
Non-negativity
Triangle inequality

In a normed space, the norm of the zero vector is:

0
1
2
3

Which of the following is an example of a normed space?

The set of all continuous functions on the interval [0, 1]
The set of all polynomials with real coefficients
The set of all vectors in $\mathbb{R}^n$
The set of all matrices with real entries

The triangle inequality in a normed space states that:

$|x + y| \leq |x| + |y|$
$|x + y| \geq |x| + |y|$
$|x + y| = |x| + |y|$
$|x + y| \leq |x| - |y|$

Which of the following is not a property of a Banach space?

Completeness
Linearity
Non-negativity
Triangle inequality

In a Banach space, the Cauchy sequence is:

A sequence that converges to a limit in the space
A sequence that is bounded in the space
A sequence that is increasing in the space
A sequence that is decreasing in the space

Which of the following is an example of a Banach space?

The set of all continuous functions on the interval [0, 1]
The set of all polynomials with real coefficients
The set of all vectors in $\mathbb{R}^n$
The set of all matrices with real entries

The completeness of a normed space means that:

Every Cauchy sequence in the space converges to a limit in the space
Every bounded sequence in the space converges to a limit in the space
Every increasing sequence in the space converges to a limit in the space
Every decreasing sequence in the space converges to a limit in the space

Which of the following is not a property of a Hilbert space?

Completeness
Inner product
Linearity
Triangle inequality

In a Hilbert space, the inner product of two vectors is:

A complex number
A real number
A vector
A matrix

Which of the following is an example of a Hilbert space?

The set of all continuous functions on the interval [0, 1]
The set of all polynomials with real coefficients
The set of all vectors in $\mathbb{R}^n$
The set of all matrices with real entries

The inner product in a Hilbert space satisfies the following properties:

Linearity, positivity, and symmetry
Linearity, non-negativity, and symmetry
Linearity, positivity, and anti-symmetry
Linearity, non-negativity, and anti-symmetry

Which of the following is not a property of a Banach algebra?

Associativity
Commutativity
Completeness
Identity element

In a Banach algebra, the norm of the product of two elements is:

Less than or equal to the product of the norms of the elements
Greater than or equal to the product of the norms of the elements
Equal to the product of the norms of the elements
None of the above

Which of the following is an example of a Banach algebra?

The set of all continuous functions on the interval [0, 1]
The set of all polynomials with real coefficients
The set of all vectors in $\mathbb{R}^n$
The set of all matrices with real entries

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