\(\ell^\infty\) is the complex vector space of bounded sequences of complex numbers with entrywise addition and scalar multiplication. Verify that the “infinity norm”
Prove that the parallelogram law does not hold for \(\norm{\cdot}_\infty\) on the space \(C[0,1]\text{.}\) Conclude that there is no inner product on \(C[0,1]\) such that \(\ip{f}{f}^{1/2} = \norm{f}_\infty\text{.}\)
is a closed linear subspace of \(RH^2\) (see Table ).
5.
Show that the subspace of polynomials is not closed in \(C[0,1]\) in the supremum norm \(\norm{\cdot}_\infty\) or in the inner product (2.1.1).
6.
\(c_0\) denotes the subspace of \(\ell^\infty\) consisting of all sequences \((x_n)\) which tend to 0 as \(n \to \infty\text{.}\) Prove that \(c_0\) is closed in \(\ell^\infty\) with respect to \(\norm{\cdot}_\infty\text{.}\)
7.
Guess the closed linear span of the set \(\{1, x, x^2, \ldots\}\) in \(C[0,1]\text{.}\) (We’ll see the answer later. This result is fundamental.)
8.
The open ball (of radius 1) in a normed space \((V, \norm{\cdot})\) is the set
\begin{equation*}
G = \{x \in V: \norm{x} \lt 1\}.
\end{equation*}
Likewise, the closed ball is
\begin{equation*}
F = \{x \in V: \norm{x} \leq 1\}.
\end{equation*}
Show that the closed ball is the closure of the open ball. (Then argue that the ball of radius \(r\) centered at \(a\) satisfies the same property.)
9.
Show that \(c_0\) is the closed linear span in \(\ell^\infty\) of the set \(\{e_n: n \in \mathbb{N}\}\text{,}\) where \(e_n\) is the sequence with a \(1\) in the \(n\)th position and \(0\) otherwise.
10.
Give an example of an inner product space \(V\) and a dense subspace \(E\) of \(V\) which is of codimension \(1\) in \(V\text{.}\)
11.
Prove that every finite dimensional subspace of a normed space is closed.
