Let \(V\) be the (real) vector space of all functions \(f\) from \(\RR\) into \(\RR\). Which of the following sets of functions are subspaces of \(V\)?
all \(f\) such that \(f(x^2) = f(x)^2\);
all \(f\) such that \(f(0) = f(1)\);
all \(f\) such that \(f(3) = 1 + f(-5)\);
all \(f\) such that \(f(-1) = 0\);
all \(f\) which are continuous.
Solution
1. A non-empty subset \(W \subseteq V\) is a subspace if and only if it contains the zero function and is closed under addition and scalar multiplication.
subspace criterion for vector spaces.
2. Part (a): The set \(W_a = \{f \in V : f(x^2) = f(x)^2\}\) is not a subspace.
2.1. Consider the identity function \(f(x) = x\).
definition.
2.2. \(f(x^2) = x^2 = (x)^2 = f(x)^2\), so \(f \in W_a\).
2.1.
2.3. Consider the scalar multiple \(g = 2f\), so \(g(x) = 2x\).
scalar multiplication in \(V\).
2.4. \(g(x^2) = 2x^2\), whereas \(g(x)^2 = (2x)^2 = 4x^2\).
direct computation.
2.5. For \(x = 1\), \(g(1^2) = 2 \neq 4 = g(1)^2\), so \(2f \notin W_a\).
4 evaluated at \(x = 1\).
2.6. Thus \(W_a\) is not closed under scalar multiplication, so it is not a subspace.
5.
3. Part (b): The set \(W_b = \{f \in V : f(0) = f(1)\}\) is a subspace.
3.1. Define the linear functional \(T: V \to \mathbb{R}\) by \(T(f) = f(0) - f(1)\).
\(T(f + g) = (f+g)(0) - (f+g)(1) = T(f) + T(g)\) and \(T(cf) = cT(f)\).
3.2. \(W_b = \ker(T) = \{f \in V : T(f) = 0\}\).
\(f(0) = f(1) \iff f(0) - f(1) = 0\).
3.3. The kernel of any linear transformation is a subspace.
standard linear algebra.
3.4. Hence \(W_b\) is a subspace of \(V\).
2 and 3.
4. Part (c): The set \(W_c = \{f \in V : f(3) = 1 + f(-5)\}\) is not a subspace.
4.1. The zero function \(0_V(x) = 0\) for all \(x \in \mathbb{R}\).
definition of the zero vector in \(V\).
4.2. \(0_V(3) = 0\), whereas \(1 + 0_V(-5) = 1 + 0 = 1 \neq 0\).
arithmetic.
4.3. Thus \(0_V \notin W_c\), so \(W_c\) is not a subspace.
every subspace must contain the zero vector.
5. Part (d): The set \(W_d = \{f \in V : f(-1) = 0\}\) is a subspace.
5.1. Define the evaluation functional \(\operatorname{ev}_{-1}: V \to \mathbb{R}\) by \(\operatorname{ev}_{-1}(f) = f(-1)\).
\(\operatorname{ev}_{-1}\) is a linear functional on \(V\).
5.2. \(W_d = \ker(\operatorname{ev}_{-1})\).
definition of \(W_d\).
5.3. The kernel of a linear map is a subspace, so \(W_d\) is a subspace of \(V\).
1 and 2.
6. Part (e): The set \(W_e = \{f \in V : f \text{ is continuous}\}\) is a subspace.
6.1. The zero function \(0_V\) is constant, hence continuous, so \(0_V \in W_e\).
constant functions are continuous.
6.2. If \(f, g \in W_e\), then \(f + g\) is continuous, so \(f + g \in W_e\).
sum of continuous functions is continuous.
6.3. If \(f \in W_e\) and \(c \in \mathbb{R}\), then \(cf\) is continuous, so \(cf \in W_e\).
product of a constant and a continuous function is continuous.
6.4. Thus \(W_e = C(\mathbb{R})\) is a subspace of \(V\).
<2>1–<2>3.
7. Conclusion: The subsets (b), (d), and (e) are subspaces of \(V\), while (a) and (c) are not.
2 through 6.
Q.E.D.