15.5 Sequence Spaces
Definition 15.5.1 ($c_{0}$-Direct Sum).label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$. For any $x \in \prod_{i \in I}X_{i}$, $x$ vanishes at infinity if for each $\eps > 0$, $\bracs{i \in I| \norm{x_i}_{X_i} \ge \eps}$ is finite. The space
equipped with the uniform norm
is the $c_{0}$-direct sum of $\seqi{X}$.
Definition 15.5.2 ($l^{p}$-Direct Sum).label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$, then the $l^{p}$-direct sum of $\seqi{X}$ is the space
equipped with the norm
Definition 15.5.3 ($l^{\infty}$-Direct Product).label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$, then the $l^{\infty}$-direct product of $\seqi{X}$ is the space
equipped with the norm
Proposition 15.5.4.label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty]$.
- (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)
(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)
(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)
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)
If $X_{i}$ is a Banach space for all $i \in I$, then so is $[l^{p}(I); X_{i}]$.
Proof. (1), (2), (3): By the classical Hölder’s inequality, Minkowski’s inequality, and Markov’s inequality.
(4): For each $i \in I$, $\norm{x_i}_{X_i}\le \norm{x}_{[l^p(I); X_i]}$, so the result holds when $q = \infty$.
If $q < \infty$, then by Proposition 15.1.12, there exists $\lambda \in [p, q]$ such that
$\square$
Theorem 15.5.5.label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$. For each $y \in [l^{1}(I); X_{i}^{*}]$, let
then the mapping
is an isometric isomorphism.
Proof. By Hölder’s Inequality, for each $y \in [l^{1}(I); X_{i}^{*}]$, $\norm{\phi_y}_{[c_0(I); X_i]^*}\le \norm{y}_{[l^1(I); X_i^*]}$.
Let $\phi \in [c_{0}(I); X_{i}]^{*}$, then there exists $y \in [l^{\infty}(I); X_{i}^{*}]$ such that for each $i \in I$ and $x_{i} \in X_{i}$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[c_0(I); X_i]}= \dpn{x_i, y_i}{X_i}$.
Let $J \subset I$ be finite and $\alpha \in (0, 1)$, then there exists $x \in [c_{0}(I); X_{i}]$ such that
- (1)
$\{i \in I|x_{i} \ne 0\} \subset J$.
- (2)
$\norm{x}_{[c_0(I); X_i]}\le 1$.
- (3)
For each $j \in J$, $\dpn{x_j, y_j}{X_j}\ge \alpha\norm{y_j}_{X_j^*}$.
Thus
As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^{1}(I); X_{i}^{*}]$ with $\norm{y}_{[l^1(I); X_i^*]}\le \norm{\phi}_{[c_0(I); X_i]^*}$. By the Dominated Convergence Theorem, $\dpn{x, \phi_y}{[c_0(I); X_i]}= \dpn{x, \phi}{[c_0(I); X_i]}$ for all $x \in [c_{0}(I); X_{i}]$. Hence the map is an isometric isomorphism.$\square$
Theorem 15.5.6.label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$ and $q \in (1, \infty]$ be Hölder conjugates. For each $y \in [l^{q}(I); X_{i}^{*}]$, let
then the mapping
is an isometric isomorphism.
Proof. Let $\phi \in [l^{p}(I); X_{i}]^{*}$, then there exists $y \in [l^{\infty}(I); X_{i}^{*}]$ such that for each $i \in I$ and $x_{i} \in X_{i}$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]}= \dpn{x_i, y_i}{X_i}$.
Since the $q = \infty$ case has been ruled out, assume that $q \in (1, \infty)$. For each $\alpha \in (0, 1)$, there exists $x \in [l^{\infty}(I); X_{i}]$ with $\norm{x_i}_{X_i}\le 1$ and $\dpn{x_i, y_i}{X_i}\ge \alpha \norm{y_i}_{X_i^*}$. For each $J \subset I$ finite and $i \in I$, let $F_{J}(i) = \one_{J}(i) \cdot \norm{y_i}_{X_i^*}^{q - 1}$, then by Lemma 15.1.4,
so
As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^{q}(I); X_{i}^{*}]$ with $\norm{y}_{[l^q(I); X_i^*]}= \norm{\phi}_{[l^p(I); X_i]^*}$.$\square$
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