functional analysis iv

Remembering: What is a completed normed linear space?

• A space with a norm and a linear structure that is complete
• A space with a norm and a linear structure that is incomplete
• A space with a linear structure but no norm
• A space with a norm but no linear structure

Understanding: What does it mean for a space to be complete?

• Every sequence has a limit
• Every set has a largest element
• Every infinite set has a limit
• Every sequence is finite

Applying: Which of the following is an example of a completed normed linear space?

• The set of polynomials with the supremum norm
• The set of rational numbers with the Euclidean norm
• The set of real numbers with the absolute value norm
• The set of continuous functions with the supremum norm

Analyzing: What is the relationship between a norm and a metric?

• A norm is a metric and a metric is a norm
• A norm is not a metric but a metric can be a norm
• A norm is a metric but a metric is not a norm
• None of the above

Evaluating: How does the completeness of a normed linear space affect its properties?

• It makes the space into a Banach space
• It makes the space into a Hilbert space
• It makes the space into a finite dimensional space
• It makes the space into a discrete space

Creating: Can a normed linear space be both complete and not complete at the same time?

• No, a space cannot be both complete and not complete at the same time
• Yes, a space can be both complete and not complete at the same time
• Yes, but only if it is not a linear space
• No, a linear space cannot be both complete and not complete at the same time