**Speaker: **Babhrubahan Bose

**Date: **16th October, 2018

**Time: **9:15pm-10:15pm

**Abstract:**

In functional analysis, we often encounter characterization of Hilbert spaces. Two nice examples are: 1) when parallelogram law holds in a Banach space and 2) when given any two vectors of the same norm in the Banach space, there exists an isometry of the space that takes one of the points to the other. Here we are asking the next natural question, i.e., when does a vector space, equipped with a metric, becomes normed linear. It is elementary to note that if the metric is translation invariant and scaling equivariant, the metric comes from a norm. In the paper that I shall present, Peter Semrl proves that if the metric is translation invariant and every one-dimensional subspace of the vector space has and isometry with the reals, then the vector space is normed linear if the dimension is 2 or more.

**Prerequisites:**Familiarity with vector spaces and a little bit of metric spaces (though not necessary). The talk will be self-contained.