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The first problem session will be held on Thursday, September 9. See the Detailed teaching plan for more details.

Tillitsmann/Trustee for the course is

Gabriel Balaban,

gabrib at math.uio.no

Exam in MAT4340 this fall will be held as a written exam.

Example. On the set R of all real numbers we define a metric d by

d(x,y)=1 if x≠y, d(x,y)=0 if x=y.

Check that d is translation invariant on R but that d fails to be homogeneous w.r.t. (scalar) multiplication. The open unit ball centered at any x in (R,d) is {x} and the closed unit ball of x is R. The closure of {x} is {x}, which is different from R. The topology of (R,d) is discrete and is not homéomorphic to the vector space topology on R induced by the standard norm (or by any norm).

A metric d on a vector space X is called translation invariant if

d(x+u,y+u)=d(x,y), for all x,y,u in X.

The metric d is said to be homogeneous if

d(ax,ay)=|a| d(x,y), for all x,y in X, and for all a in the field F.

Remark. If a metric d is both translation invariant and homogeneous on X, we may define a norm by

||x||=d(x,0), for all x in X.

You might wish to verify that this really gives a norm on X, and that d(...

The first lecture will be given on Tuesday, August 24. We will give a brief review of metric spaces and normed spaces with some examples (sections 1.2, 1.3, 2.1, and 2.2).