# Proving that $X$ is a Banach Space and $Y$ is Not

• MHB
• mathmari

## 1. What is a Banach space?

A Banach space is a complete normed vector space. This means that it is a vector space (a set of elements that can be added and multiplied by scalars) with a norm (a mathematical concept that measures the length or size of a vector) that satisfies the completeness property (every Cauchy sequence converges to a point in the space).

## 2. How do you prove that a space is Banach?

To prove that a space is Banach, you must show that it is a complete normed vector space. This can be done by showing that it satisfies the three properties of a normed vector space: vector addition, scalar multiplication, and the triangle inequality. Then, you must also show that it satisfies the completeness property by proving that every Cauchy sequence in the space converges to a point in the space.

## 3. What is an example of a Banach space?

The most commonly known example of a Banach space is the space of real or complex numbers, with the norm defined as the absolute value or modulus. Another example is the space of continuous functions on a closed interval, with the norm defined as the maximum absolute value of the function.

## 4. How do you prove that a space is not Banach?

To prove that a space is not Banach, you must show that it fails to satisfy one or more of the properties of a complete normed vector space. This could be due to a lack of completeness, a violation of the triangle inequality, or a failure to satisfy the properties of vector addition and scalar multiplication.

## 5. Can a subspace of a Banach space also be Banach?

Yes, a subspace of a Banach space can also be Banach. This is because a subspace inherits the properties of its parent space, including completeness and normed vector space properties. However, it is possible for a subspace to be incomplete or not satisfy the properties of a normed vector space, in which case it would not be considered a Banach space.

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