Exercise 6.3.24

Answers

1.

Check

\begin{array}{l} T (cσ + τ) (k) & = \sum_{i = k}^{\infty} (cσ + τ) (k) \\ = c \sum_{i = k}^{\infty} (σ) (k) + \sum_{i = k}^{\infty} (τ) (k) \\ = cT (σ) (k) + T (τ) (k) . \end{array}

2.

For

k \leq n

we have

T (e_{n}) (k) = \sum_{i = k}^{\infty} e_{n} (i) = 1 = \sum_{i = 1}^{n} e_{i} (k) .

And for $k > n$ we have

T (e_{n}) (k) = \sum_{i = k}^{\infty} e_{n} (i) = 0 = \sum_{i = 1}^{n} e_{i} (k) .

3.

Suppose that

T^{∗}

exist. We try to compute

T^{∗} (e_{1})

1 \cdot \bar{T^{∗} (e_{1})} (i) = ⟨ e_{i}, T^{∗} (e_{1}) ⟩ = ⟨ \sum_{i = 1}^{n} e_{i}, e_{1} ⟩ = 1 .

This means that $T^{∗} (e_{1}) (i) = 1$ for all $i$ . This is impossible since $T^{∗} (e_{1})$ is not an element in $V$ .

jephianlin

2011-06-27 00:00