Jamma said:I think that it is automatically continuous. Is that what you wanted instead? As noted, this clearly doesn't imply linearity.
mathplease said:i see, yes. so in general, to show that something is a bounded linear operator from X to Y, you need to show the inequality, prove linearity and show that its a mapping from X to Y?
micromass said:Yes, these are the things you need to show!
A bounded operator is a linear transformation between two normed vector spaces that preserves the structure of the vector space and satisfies a certain boundedness condition. This means that the operator does not "blow up" or become unbounded when applied to elements of the vector space.
Linearity refers to the property of a bounded operator to preserve the properties of addition and scalar multiplication. In other words, a bounded operator is linear if it satisfies the following conditions: f(x + y) = f(x) + f(y) and f(cx) = cf(x), where x and y are elements of the vector space and c is a scalar.
A bounded operator is determined by its norm, which is the supremum (least upper bound) of the operator's values for all elements in the vector space. If the norm is finite, the operator is considered bounded. If the norm is infinite, the operator is considered unbounded.
Inequalities are used to describe the boundedness of operators. An operator is considered bounded if its norm is less than or equal to a constant value. This constant value is referred to as the operator's bound. Inequalities are also used to compare the norms of different operators.
Bounded operators are fundamental in functional analysis and are used to study linear transformations between different vector spaces. They also play a key role in the study of differential equations and other areas of mathematics where linear operators are used to model real-world problems.