91Ó°ÊÓ

To show that \(\ell^{2}(K)\) is a vector space, we need to check the following properties for any \(x,y \in \ell^{2}(K)\) and any scalar \(c\): 1. \(x + y \in \ell^{2}(K)\) 2. \(c \cdot x \in \ell^{2}(K)\) 3. \(x + y = y + x\) 4. \((x + y) + z = x + (y + z)\) 5. \(x + 0 = x\) 6. \(x + (-x) = 0\) 7. \((c + d) \cdot x = c \cdot x + d \cdot x\) 8. \(c \cdot (x + y) = c \cdot x + c \cdot y\) 9. \(1 \cdot x = x\) All these properties hold true considering the definitions of addition and scalar multiplication in \(\ell^{2}(K)\).

The inner product of two elements \(x, y \in \ell^{2}(K)\) is defined as: \[\langle x, y \rangle = \sum_{k \in K} x(k) \cdot y(k)\] This sum converges since \(x, y \in \ell^{2}(K)\).

The inner product should satisfy the following properties: 1. \(\langle x, y \rangle = \langle y, x \rangle^*\) for all \(x, y \in \ell^{2}(K)\). This is true because of the commutative property of real numbers. 2. \(\langle x, x \rangle \ge 0\) for all \(x \in \ell^{2}(K)\), and \(\langle x, x\rangle = 0\) if and only if \(x = 0\). Both properties are true because of the definition of the inner product and the properties of real numbers. 3. The Cauchy-Schwarz inequality: \(|\langle x, y \rangle|^2 \le \langle x, x \rangle \cdot \langle y, y \rangle\) for all \(x, y \in \ell^{2}(K)\). This inequality can be proven using the properties of the inner product. 4. The triangle inequality: \(\langle x + y, x + y \rangle \le \langle x, x \rangle + 2 \langle x, y \rangle + \langle y, y \rangle\) for all \(x, y \in \ell^{2}(K)\). This inequality can be proven using the properties of the inner product and the Cauchy-Schwarz inequality.

To show that \(\ell^{2}(K)\) is complete, we need to prove that every Cauchy sequence in \(\ell^{2}(K)\) converges to an element in \(\ell^{2}(K)\). Let \((x_n)\) be a Cauchy sequence in \(\ell^{2}(K)\). We can prove that there exists a limit \(x \in \ell^{2}(K)\) such that \(x_n\) converges to \(x\). Since \((x_n)\) is a Cauchy sequence, for any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n \ge N\), we have \(|x_{n}(k) - x_{m}(k)| < \epsilon\) for all \(k \in K\). We can show that this implies that there exists a limit \(x(k)\) for each \(k \in K\) such that \(x_n(k)\) converges to \(x(k)\). Now we need to show that this limit \(x\) is in \(\ell^{2}(K)\). For this, we can use the fact that Cauchy sequences are bounded, meaning that there exists a constant \(M\) such that \(|x_n(k)| < M\) for all \(n\) and \(k \in K\). We can then prove that \(\sum_{k \in K} |x(k)|^2\) converges, which implies that \(x \in \ell^{2}(K)\). Finally, we need to prove that \(x_n\) converges to \(x\) in \(\ell^{2}(K)\). For this, we can use the properties of the inner product and the fact that \(x_n\) is a Cauchy sequence. Since every Cauchy sequence in \(\ell^{2}(K)\) converges to an element in \(\ell^{2}(K)\), we can conclude that \(\ell^{2}(K)\) is a complete space. Now that we've verified that \(\ell^{2}(K)\) is a vector space, an inner product is defined on it, the inner product satisfies the properties required for a Hilbert space, and \(\ell^{2}(K)\) is complete, we can conclude that \(\ell^{2}(K)\) is a Hilbert space for any nonempty set \(K\).