J. Horvath (1966), Topological Vector--Spaces and Distributions, Addison-Wesley Series in Math.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....three properties: ffl Addition and scalar multiplication are continuous, when the field k is given the discrete topology. ffl is hausdorff. ffl 0 2 V has a neighborhood basis of open linear subspaces. The first requirement means that we have a topological vector space in the sense of [24] (except that most texts take the field to be the real or complex numbers with its usual topology) The third requirement is quite stringent. For example, it implies that the only linear topology on a finite dimensional vector space is the discrete topology. Let T VEC denote the category whose ....

.... = V Omega W Thus we cannot hope for any kind of completeness theorem. RT VEC does not satisfy such an identity. This point is discussed in [10] 5.2 Quotients and Direct Sums We now discuss quotients and direct sums of topological vector spaces. More complete discussions can be found in [24] and [41] Given a topological vector space V and an arbitrary linear subspace U , it is readily seen that the quotient topology on the quotient space V=U gives a topological vector space. It is not generally the case however that when an object of T VEC is quotiented by an arbitrary subspace ....

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J. Horvath, Topological Vector Spaces and Distributions, Addison-Wesley Series in Mathematics, (1966).

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J. Horvath (1966), Topological Vector--Spaces and Distributions, Addison-Wesley Series in Math.

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