## The space of real vector measures is complete

Let be a measure space. For a real vector measure we define its *total variation * by

.

Prove that the space of vector measures endowed with is a Banach space.

## A lemma of J. L. Lions

Let and be three Banach spaces with norms and . Assume that with *compact *injection and that with *continuous *injection. Prove that

satisfying .

*Applications:*

- Prove that for every there exists satisfying.
- Pick . Prove that for every there exists such that .

*Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011*

## Interesting inequality involving Banach spaces and an operator

Let be two Banach spaces with norms . Let (space of linear bounded operators ) be such that is closed and . Let denote another norm on which is weaker than , i.e. .

Prove that there exists a constant such that .

Haim Brezis,* Functional Analysis, Sobolev Spaces and Partial Differential Equations*, Chapter 2

## Conditions for an operator to be bounded

Let be a Banach space, and let be a linear operator. In each of the following cases, prove that is bounded.

a) ;

b) .

## Bilinear Continuous Operator

Let be three Banach spaces and consider a bilinear operator . Prove that is continuous if and only if there exists a constant such that .

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## non-Separable space Example 1

Prove that the space is not separable.

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## Separable spaces

A Banach space is called separable if it contains a countable dense subset. Here are some interesting facts about separable spaces.

Prove that if is a subset of a separable metric space then is also separable.

If the dual of a normed vector space is separable, then the space itself is separable.