In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous. is continuous at some point is continuous at the origin in If is locally convex then this list may be extended to include: for every continuous seminorm on there exists a continuous seminorm on such that If and are both Hausdorff locally convex spaces then this list may be extended to include: is weakly continuous and its transpose maps equicontinuous subsets of to equicontinuous subsets of If is a sequential space (such as a pseudometrizable space) then this list may be extended to include: is sequentially continuous at some (or equivalently, at every) point of its domain. If is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list: is a bounded linear operator (that is, it maps bounded subsets of to bounded subsets of ). If is seminormable space (such as a normed space) then this list may be extended to include: maps some neighborhood of 0 to a bounded subset of If and are both normed or seminormed spaces (with both seminorms denoted by ) then this list may be extended to include: for every there exists some such that If and are Hausdorff locally convex spaces with finite-dimensional then this list may be extended to include: the graph of is closed in Throughout, is a linear map between topological vector spaces (TVSs). Bounded subset Bounded set (topological vector space) The notion of a "bounded set" for a topological vector space is that of being a von Neumann bounded set.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.