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    Contraction of Tensors

    To George: Aha, ok, if you put it this way then I agree that it becomes a bit less controversial…but still I would then feel inclined to relabel the indices on initial step and then it would actually look all the same, but with relabeled indices. Alright, perhaps I just need a bit more practice...
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    Contraction of Tensors

    Hi all! I've got a short question concerning a minor notational issue about tensor contraction I've run across recently. Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor. Then their total contraction is zero: C_1^1C_2^2\,A \otimes S=0. As a proof one simply computes...
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    Not every metric comes from a norm

    this was easier than I thought. In fact take x=(1,1,...), then 2||x|| is not equal to ||2x||! First I tried to see whether the triangle inequality is not satisfied, but it did not to work, because both the norm and the metric seem somehow to be "conform" in this respect.
  4. L

    Not every metric comes from a norm

    As a matter of fact I know what a norm is.
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    Not every metric comes from a norm

    Hello! It is said that not every metric comes from a norm. Consider for example a metric defined on all sequences of real numbers with the metric: d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|} I can't grasp how can that be. There is a proof...
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    Subbasis help

    Hello, could you please check if the reasoning is correct. This is not a homework, just a part of an exercise in a book I'm reading at the moment. Suppose X is a set, \mathcal{B}:=\{S\subset{}X:\bigcup{}S=X\}, \\...
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    Basic lemma in topology

    Ups, of course I meant r=|b-a|. Sorry for that dumb misprint! :smile: Thanks!
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    Basic lemma in topology

    Hello, there is a basic lemma in topology, saying that: Let X be a topological space, and B is a collection of open subsets of X. If every open subset of X satisfies the basis criterion with respect to B (in the sense, that every element x of an open set O is in a basis open set S, contained...
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    Free homotopy

    Hello, Here is a short lemma: A path-connected space X is simply-connected iff any two loops in X are free homotopic. My question is whether it is allowed to use a straight-line homotopy straight away in order to construct a free homotopy? For example, let u and v be two loops and w is a...
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