Sentence examples for bounded linear functionals on from inspiring English sources
for every ; are positive bounded linear functionals on given by (2.23).
By (X^), we denote the space of all bounded linear functionals on X. (X^) is called the continuous dual of a normed space X.
We now use the fact that 〈 ⋅, k 〉 and 〈 ⋅, α J k 〉 are bounded linear functionals on K together with (2.7) to rewrite the above inequality as 2 〈 k, k 〉 K − 〈 P k, k 〉 K − 〈 k, P k 〉 K ≤ − 〈 P k, α J k 〉 K − 〈 α J k, P k 〉 K, (3.4).
In particular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter.
If (e_{w}) is a bounded linear functional on H, then the Riesz representation theorem implies that there is a function (usually denoted (K_{w})) in H that induces this linear functional, that is, (e_{w}(f)=langle f,K_{w}rangle).
This inequality also shows that the point evaluation: ev z ( f ) : = f ( z ), z ∈ D is a bounded linear functional on the Bergman space (take K : = { z } ).
The following result implies that convergence in (aleph_{alpha } ^{p} -norm implies the uniform convergence on each compact subset of (mathcal{C}_{n}(Gamma )) and point evaluation is a bounded linear functional on (aleph_{alpha }^{p} -norm
Any (varphi in L^2) acts as a bounded linear functional on X via (ell _varphi (psi ) = langle bar{varphi },psi rangle ) so (L^2 subset X^*) which can be shown to be dense.
Based on this fact, Bownik [31] (Theorem 2) constructed a surprising example of a linear functional defined on a dense subspace of H 1 ( R n ), which maps all ( 1, ∞, 0 ) -atoms into bounded scalars, but yet cannot extend to a bounded linear functional on the whole H 1 ( R n ).
