Definition 17.1.3.label Let $(E_{0}, E_{1})$ be a compatible couple of normed vector spaces over $K \in \RC$, then:
- (1)
$E_{0} \cap E_{1}$ is a normed space under the norm
\[\norm{\cdot}_{E_0 \cap E_1}: E_{0} \cap E_{1} \to [0, \infty) \quad x \mapsto \max(\norm{x}_{E_0}, \norm{x}_{E_1})\] - (2)
$E_{0} + E_{1}$ is a normed space under the norm
\[\norm{\cdot}_{E_0 + E_1}: E_{0} + E_{1} \to [0, \infty)\]with
\[x \mapsto \inf\bracsn{\norm{x_0}_{E_0} + \norm{x_1}_{E_1}|x_0 \in E_0, x_1 \in E_1, x = x_0 + x_1}\] - (3)
If $E_{0}$ and $E_{1}$ are Banach spaces, then $E_{0} \cap E_{1}$ and $E_{0} + E_{1}$ are also Banach spaces.
The norms on $E_{0} \cap E_{1}$ and $E_{0} + E_{1}$ defined above are the intersection and sum norms of the couple, respectively.
Proof. (2): Let $x, y \in E_{0} + E_{1}$, $x_{0}, y_{0} \in E_{0}$, $x_{1}, y_{1} \in E_{1}$ such that $x = x_{0} + x_{1}$ and $y = y_{0} + y_{1}$, then
As this holds for all choices of $x_{0}, y_{0} \in E_{0}$ and $x_{1}, y_{1} \in E_{1}$,
(3): Let $\seq{x_n}\subset E_{0} + E_{1}$ such that $\sum_{n \in \natp}\norm{x_n}_{E_0 + E_1}< \infty$. For each $n \in \natp$, let $y_{n} \in E_{0}$ and $z_{n} \in E_{1}$ with $x_{n} = y_{n} + z_{n}$ and $\norm{y_n}_{E_0}+ \norm{z_n}_{E_1}\le 2\norm{x_n}_{E_0 + E_1}$. Since $E_{0}$ and $E_{1}$ are both complete, $y = \sum_{n = 1}^{\infty} y_{n}$ exists in $E_{0}$ and $z = \sum_{n = 1}^{\infty} z_{n}$ exists in $E_{1}$. Let $x = y + z$, then for each $N \in \natp$,
as $N \to \infty$. Therefore $E_{0} + E_{1}$ is also a Banach space.$\square$
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