Discussion
Consider the statements
P: If \(X\) is a normed linear space and \(M \subseteq X\) is a subspace, then the closure \(\bar{M}\) is also a subspace of \(X\).
Q: If \(X\) is a Banach space and \(\sum x_{n}\) is an absolutely convergent series in \(X\), then \(\sum x_{n}\) is convergent.
R: Let \(M_{1}\) and \(M_{2}\) be subspaces of an inner product space such that \(M_{1} \cap M_{2} = \{0\}\). Then \(\forall m_{1} \in M_{1},\ m_{2} \in M_{2}\); \(\|m_{1} + m_{2}\|^{2} = \|m_{1}\|^{2} + \|m_{2}\|^{2}\).
S: Let \(f : X \to Y\) be a linear transformation from the Banach Space \(X\) into the Banach space \(Y\). If \(f\) is continuous, then the graph of \(f\) is always compact.
The correct statements amongst the above are:
Answer: C
The correct statements are P and Q only.
P is true: if \(x_{n} \to x\) and \(y_{n} \to y\) with \(x_{n}, y_{n} \in M\), then \(\alpha x_{n} + \beta y_{n} \to \alpha x + \beta y\), so the closure of a subspace is a subspace. Q is true and in fact characterises Banach spaces: the partial sums of an absolutely convergent series form a Cauchy sequence, which converges by completeness.
R is false: the Pythagorean identity needs \(M_{1} \perp M_{2}\), not merely \(M_{1} \cap M_{2} = \{0\}\). In \(\mathbb{R}^{2}\), with \(m_{1} = (1, 0)\) and \(m_{2} = (1, 1)\) on two different lines, \(\|m_{1} + m_{2}\|^{2} = 5\) but \(\|m_{1}\|^{2} + \|m_{2}\|^{2} = 3\). S is false: the graph of a continuous linear map is closed, but it contains \((tx, tf(x))\) for every scalar \(t\), so it is unbounded and never compact unless \(X = \{0\}\).
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