In this bachelor's thesis, the proof of Dugundji’s extension of Tietze extension theorem for mappings from a closed subset of a metric space into a locally convex linear space is presented in detail together with its immediate consequences. The theorem where a paracompact Hausdorff space is considered instead of a metric one is also proved. The Dugundji extension property concerning the existence of a simultaneous extender is defined and it is proved that not only metric but also stratifiable spaces have this property. With the example of the Sorgenfrey line it is shown that it is sometimes possible to find such simultaneous extenders in spaces that are not stratifiable, while with the example of the Michael line it is shown that not all normal spaces have this property, even if they are paracompact Hausdorff spaces.
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