By Abraham R., Marsden J.E.
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X E,+F is multilinear iff it is linear in each variable separately. Note that this does not mean f is linear on the product vector space. 4 Theorem. For finite-dimensional real vector spaces, linear and multilinear maps are continuous. 3. The proof is a consequence of DieudonnC [1960, p. 991. The following is an immediate corollary of this, but is also true more generally (DieudonnC [1960, p. 891). 5 Corollary. Addition and scalar multiplication in a (normed) vector space are continuous maps from E X E + = E and R x E+E, respectively.
If cp(0)= u, g,(l) = v , we say cp joins u and v ; S is called arcwise connected iff every two points in S can be joined by an arc in S. A space is called locally arcwise connected iff each point has an arcwise connected neighborhood (in the relative topology). The relationship with connectedness is the following. 33 Proposition. Every arcwise connected space is connected. If a space is connected and locally arcwise connected, it is arcwise connected. In particular, a space locally homeomorphic to R nis connected iff it is arcwise connected.
For example, the rationals are not complete relative to the absolute value norm. For the necessity of finite dimension, the space of continuous functions on [0, 11 has two inequivalent norms (DieudonnC [1960, p. 1021). 3 tells us that a unique topology is determined by norms. lll(f (eo)k where Recall that f: El x E, x , . x E,+F is multilinear iff it is linear in each variable separately. Note that this does not mean f is linear on the product vector space. 4 Theorem. For finite-dimensional real vector spaces, linear and multilinear maps are continuous.