Compact operator maps weakly convergent sequences into strong convergent sequences

Suppose is a compact operator and is a sequence in such that (i.e. converges weakly). Prove that strongly in .

Slick condition for boundedness of an operator

Consider a Banach space, and the space of real functionals on . Suppose that is a linear operator such that for all . Prove that is bounded.

Compact & Hausdorff spaces

The following problem wants to prove that compact and Hausdorff spaces have an interesting property. If is Hausdorff and compact, and we consider another Hausdorff topology coarser than and Hausdorff, then .

1) Prove that a compact set in a Hausdorff space is closed.

2) Every continuous function from a compact space onto a Hausdorff space is open.

3) If you have a bijective continuous function from a compact space to a Hausdorff space, then the two spaces are homeomorphic.

A short application, which was my motivation for this post is the following problem:

Let be a Banach space and be a compact subset in the strong topology. Let be a sequence in such that in the weak topology. Prove that strongly.

Weakly Compact means bounded

Let be a Banach space, and be a compact set in the weak topology. Then is bounded.

The complement of a kernel intersection

Let be a vector space, and be linear functionals. Consider a complement of the space . Prove that is finite dimensional and its dimension is at most .

Functional is continuous iff kernel is closed

Let be a linear functional on a topological vector space . Assume for some . Then the following properties are equivalent:

1. is continuous
2. is closed
3. is not dense in
4. is bounded in some neighborhood of .

Linear maximal subspaces & Linear functionals

Take to be a vector space over the field .
1. Let be a linear functional, which is not identically zero. Then it’s kernel, , is a linear maximal subspace of .

2. Conversely, given a maximal subspace of , there exists a linear functional such that .

3. As a consequence of the above prove that if we have two linear functionals (different from zero identity) such that then there exists such that .

As application to the above solve the following:
4. Given linear functionals on such that prove that we can find scalars such that .

Prove that if are linearly independent linear functionals on then there exist elements such that , where .