Proposition 14.3.3.label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty]$.

1. (1)

   (Hölder’s Inequality) Let $q \in [1, \infty]$ be the Hölder conjugate of $p$, $\seqi{Y}, \seqi{Z}$ be normed spaces, and $\seqi{\lambda}$ such that for each $i \in I$, $\lambda \in L^{2}(X_{i}, Y_{i}; Z)$.

   For each $x \in [l^{p}(I); X_{i}]$ and $y \in [l^{q}(I); Y_{i}]$, let $\lambda(x, y)_{i} = \lambda_{i}(x_{i}, y_{i})$, then

   \[\norm{\lambda(x, y)}_{[l^1(I); Z_i]}\le \norm{x}_{[l^p(I); X_i]}\cdot \norm{y}_{[l^q(I); X_i]}\cdot \sup_{i \in I}\norm{\lambda_i}_{L^2(X_i, Y_i; Z_i)}\]

   
   
2. (2)

   (Minkowski’s Inequality) For each $x, y \in [l^{p}(I); X_{i}]$,

   \[\norm{x + y}_{[l^p(I); X_i]}\le \norm{x}_{[l^p(I); X_i]}+ \norm{y}_{[l^p(I); X_i]}\]

   
   
3. (3)

   (Markov’s Inequality) If $p < \infty$, then for each $\alpha > 0$ and $x \in [l^{p}(I); X_{i}]$,

   \[|\bracsn{i \in I|\ \norm{x}_{X_i} \ge \alpha}| \le \frac{1}{\alpha^{p}}\norm{f}_{[l^p(I); X_i]}^{p}\]

   
   

   In particular, $\bracs{i \in I|x_i \ne 0}$ is countable.

4. (4)

   For any $q \in [p, \infty]$, $[l^{p}(I); X_{i}] \subset [l^{q}(I); X_{i}]$, where for any $x \in [l^{p}(I); X_{i}]$, $\norm{x}_{[l^q(I); X_i]}\le \norm{x}_{[l^p(I); X_i]}$.

5. (5)

   If $X_{i}$ is a Banach space for all $i \in I$, then so is $[l^{p}(I); X_{i}]$.