C[a,b]上半范数的两个问题

Two problem of the subnorms on C[a,b]

设C[a,b]表示[a,b]上全体实值连续函数,它在通常函数的加法、实数与函数的数乘及函数与函数的乘法下,构成一个实数域R上无限维交换代数.研究了C[a,b]上面的半范数Np(f(x))=[b∫a|f(x)|pdx]1p(0<p<∞)与半模N∞(f(x))=maxa x b[|f(x)|]的关系,通过这种关系证明了Np(f(x))对0<p<∞不是C[a,b]上的稳定半范数.给出了C[a,b]上不是连续半模的一个实例.

Let C[ a, b] be a set of all real-valued continuous function on [ a, b]. It is an infinite dimensional commutive algebra with regard to general function operations（scalar multiplication, addition and multiplication）. A relation has been studied between subnorms Np （f（x）） and submodulus N∞ [f（x） ]. Based on this relationship, it proves that Np （f（x）） are not stable subnorms for 0 〈 p 〈 ∞ . Finally, an example of discontinuous submodules on C[ a, b] is given.