Bounded set (topological vector space)
In the context of topological vector spaces, a set is considered bounded if it is "small" in some sense, relative to the neighborhoods of zero. Several equivalent definitions exist, each capturing this intuitive notion.
Formally, let X be a topological vector space (TVS) over a field K, where K is either the real numbers ℝ or the complex numbers ℂ. A subset B of X is said to be bounded (or sometimes von Neumann bounded) if it satisfies any of the following equivalent conditions:
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For every neighborhood U of the origin (0) in X, there exists a scalar s > 0 such that B ⊆ sU = {sx : x ∈ U}. In other words, we can "inflate" any neighborhood of zero to contain the entire set B.
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For every sequence of scalars (tn) converging to 0 and every sequence (xn) in B, the sequence (tnxn) converges to 0 in X.
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For every continuous seminorm p on X, sup{p(x) : x ∈ B} < ∞. This means that the seminorm is bounded on the set B.
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For every neighborhood U of the origin in X, there exists t > 0 such that tB ⊆ U. This highlights the scaling property related to neighborhoods of zero.
These definitions are equivalent and provide different perspectives on the boundedness of a set within a topological vector space. Understanding these definitions is crucial for working with topological vector spaces and proving various results about their properties, especially concerning convergence, completeness, and duality.
In finite-dimensional topological vector spaces, a set is bounded if and only if it is contained in some large ball. However, this equivalence does not hold in infinite-dimensional spaces, where the notion of boundedness becomes more subtle and depends heavily on the topology of the vector space.