Determine which of the following conditions describe...

Determine which of the following conditions describe subspaces of the vector space $\mathrm{C}^1$ consisting of all continuously differentiable scalar functions $f(x)$. (a) $f(2)=f(3),(b) f^{\prime}(2)=f(3),(c) f^{\prime}(x)+f(x)=0,(d) \quad f(2-x)=f(x)$, (e) $f(x+2)=f(x)+2$, (f) $f(-x)=e^x f(x)$. (g) $f(x)=a+b|x|$ for some $a, b \in \mathbb{R}$,

Determine which of the following conditions describe subspaces of the vector space $\mathrm{C}^1$ consisting of all continuously differentiable scalar functions $f(x)$.
(a) $f(2)=f(3),(b) f^{\prime}(2)=f(3),(c) f^{\prime}(x)+f(x)=0,(d) \quad f(2-x)=f(x)$,
(e) $f(x+2)=f(x)+2$, (f) $f(-x)=e^x f(x)$. (g) $f(x)=a+b|x|$ for some $a, b \in \mathbb{R}$,

Key Concepts

Symmetry Conditions in Function Spaces

Symmetry or invariance conditions, such as requiring a function to be even, odd, or satisfy a relation involving transformations of the variable, are usually linear if they can be expressed as equations that preserve addition and scalar multiplication. Such conditions are important when determining subspaces defined through the symmetry properties of functions.

Affine Versus Linear Conditions

Not all function conditions are linear. Affine conditions, which impose a translation (i.e. a nonzero constant term) on the function or its evaluations, typically do not include the zero function and are not closed under scalar multiplication, and hence do not define subspaces.

Kernel of a Linear Transformation

Many conditions on functions can be interpreted as stating that the function lies in the kernel (null space) of a linear functional or operator. Since the kernel of any linear transformation is a subspace, this concept is central in determining whether sets of functions defined by linear equations form subspaces.

Linear Constraints and Homogeneity

When a condition on functions is expressed as an equation that is homogeneous (typically taking the form of a linear equation equal to zero), it defines a linear constraint. Such constraints are likely to yield subspaces because they preserve the operations of addition and scalar multiplication, and the zero function satisfies them.

Vector Subspace Criteria

A subset of a vector space is a subspace if it is non?empty and closed under both addition and scalar multiplication. In practice, this means that the zero element must be present in the subset, and that the sum of any two elements and any scalar multiple of an element also belong to the subset.

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