Normed Vector Spaces

A normed vector space is a real or complex vector space in which a norm has been defined. Formally, one says that a normed vector space is a pair (V, ∥ · ∥) where V is a vector space over K and ∥ · ∥ is a norm in V , but then one usually uses the usual abuse of language and refers to V as being the normed space. Sometimes (frequently?) one has to consider more than one norm at the same time; then one uses sub-indices on the norm symbol: ∥x∥1, for example. When dealing with several normed spaces it is also customary to refer to the norm of a space denoted by V by the symbol ∥ · ∥V . Other symbols for norms include | · | and ∥| · |∥. Exercise 1 Let (V, ∥ · ∥) be a normed vector space. Prove ∣∣∥x∥ − ∥y∥∣∣ ≤ ∥x− y∥ for all x, y ∈ V .