(X,A,u) 為measure space
Thm. 12.46
If (X,A,u) is a measure space and 1<= p <= inf then L^p(X) is a complete Banach space.
這個定理包含了Minkowski's inequality (Thm. 12.56)
Thm. 12.48
For 1<= p <= inf, simple function defined on X is dense in L^p(X,u)
這個定理可以用來證明 L^p ( 1 <= p < inf ) 是 separable
Thm. 12.49
If 1<= p < inf then L^p(X,u) is a separable metric space.
Thm. 12.50
If 1<= p < inf then Cc(R^n) is dense in L^p(R^n).
Thm. 12.51
If 1<= p < inf then Cc^inf(R^n) is dense in L^p(R^n).
利用 Approximation identity 證明 Cc^inf(R^n) is dense in Cc(R^n)
"Is has been said that analysis is the art of estimation."
Holder conjugate: p,q such that 1/p + 1/q = 1
Thm. 12.54 (Holder) Suppose f 屬於 L^p, q 屬於 L^q, then abs( integral( f g ) ) <= ||f||_p ||g||_q
For the case p = q = 2, it becomes Cauchy-Schwartz's inequality
Thm. 12.55 If (X,u) is a finite measure space, then L^1 包含於 L^p 包含於 L^q 包含於 L^inf
其中p < q
Thm. 12.58 (Young's inequality) Suppose 1/p + 1/q = 1 + 1/r. If f 屬於 L^p, g 屬於 L^q then f*g 屬於L^r and ||f*g||_r <= ||f||_p ||g||_q
For the special case p = 1, q = inf, r = inf, Young's講的就是穩定的LTI系統BIBO的特性。
從這裡也可看出對於穩定的線性系統來說，輸入訊號跟輸出訊號都屬於同個L^p。
The dual space of L^p
Thm. 12.59 If 1 < p < inf then all the linear functional D on L^p is of the form:
D(f) = integrate( f g ) for some g 屬於L^q.
For sigma-finite space, the theorem holds for p = 1 and q = inf
Moreover, ||D|| = ||g||_q
