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A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
A linear map is "bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily continuous [2] (even if its domain is not a normed space) and thus also bounded (because a continuous linear operator is always a bounded linear operator). [6]
Then a linear operator from U to V is called bounded if there exists c > 0 such that ‖ ‖ ‖ ‖ for every x in U. Bounded operators form a vector space. Bounded operators form a vector space.
A bounded linear operator T : X → Y is called completely continuous if, for every weakly convergent sequence from X, the sequence () is norm-convergent in Y (Conway 1985, §VI.3). Compact operators on a Banach space are always completely continuous.
The spectrum of is the set of all for which the operator does not have an inverse that is a bounded linear operator. Since T − λ I {\displaystyle T-\lambda I} is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem , it is bounded.
In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.
The family of finite-rank operators () on a Hilbert space form a two-sided *-ideal in (), the algebra of bounded operators on . In fact it is the minimal element among such ideals, that is, any two-sided *-ideal I {\displaystyle I} in L ( H ) {\displaystyle L(H)} must contain the finite-rank operators.
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C).