functional analysis ii

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.

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

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.

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.

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.

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

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

• Non-negativity
• Symmetry
• Triangle inequality
• Commutativity

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.