After presenting some useful inequalities on linear spaces, we introduce norms and the topologies they induce on linear spaces. Then, we discuss inner products, angles between vectors, and …

Frobenius squared all the |aij|2 and added; his norm is the square root. This treats A like a long vector with n2 components: sometimes kAkF useful, but not the choice here.

While the norm on Rn that comes from the standard inner product is the standard norm, the sup-norm on Rn does not arise from an inner product, i.e., there is no inner product whose …

So far we have shown that an inner product on a vector space always leads to a norm. The following proposition shows that we can get the inner product back if we know the norm.

Norms generalize the notion of length from Euclidean space. A norm on a vector space V is a function k k : V ! R that satis es. for all u; v 2 V and all 2 F. A vector space endowed with a …

Remark In the homework you will use Hölder’s and Minkowski’s inequalities show that the p-norm is a norm.

Key duality between SPD matrices, generalized inner products, and norms on linearly-transformed vector spaces. Not all norms come from inner products. We can have vector …