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Figure from:Weak convergence in infinite dimensional spaces

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These spaces are also infinite dimensional, the dimension can be said to be even incountable, since the segment consists of incountable many points. But they form the bases for analysis of problems in most applications e. g. in mathematical physics, mathematical modelling, optimal design etc. Let us fix a bounded interval, say J = (0,1). We shall consider functions x(¢) on this interval. The distance between functions can be measured using various norms, we shall deal with the integral norm and integral scalar product Spaces of integrable functions

Figure 1 These spaces are also infinite dimensional, the dimension can be said to be even incountable, since the segment consists of incountable many points. But they form the bases for analysis of problems in most applications e. g. in mathematical physics, mathematical modelling, optimal design etc. Let us fix a bounded interval, say J = (0,1). We shall consider functions x(¢) on this interval. The distance between functions can be measured using various norms, we shall deal with the integral norm and integral scalar product Spaces of integrable functions

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Fig. 5 Function f and the minimizing “step-slope” function. In the introduced one dimensional example the smallest value of the functional
Fig. 5 Function f and the minimizing “step-slope” function. In the introduced one dimensional example the smallest value of the functional
Thus we obtained “almost minimum”, the narrower the steps are the narrower are the side stripes and the value is closer to the minimum.
Thus we obtained “almost minimum”, the narrower the steps are the narrower are the side stripes and the value is closer to the minimum.