Which subsets are vector spaces of V?

Question

Let V be the vector space F(ℝ,ℝ). Which ones of the following subsets are vector subspaces of V ? Justify your answer! (a) {f ∈ V | f(1) = 0} (b) {f ∈ V | f(0) = 1} (c) {f ∈ V | f only has a finite number of zeros} (d) {f ∈ V | f has infinitely many zeros}

I do not know how to proceed. Could you please give me a hint? Thank you in advance.

Answer 1

A non-empty subset of a vector space is a subspace if it is closed under addition and scalar multiplication.

For example in case of (b) if $$f(0)=1$$ and $$g(0)=1,$$ then $$(f+g)(0)=2\ne 1$$ which indicates that this set is not closed under addition.

Now you can proceed and solve your problem.

Answer 2

Note that for a subspace all the following three properties must be satisfied:

$1) \ \vec{0} \in W\\ 2) \ \vec{v}+\vec{w} \in W\\ 3) \ \vec{cv}\to c \cdot \vec{v} \ ,c \in \mathbb{R}$

Then note for $f:\mathbb{R}\to\mathbb{R}$

• (a) {f ∈ V | f(1) = 0}

$$\vec 0\in V\quad f_1(1)+f_2(1)=0 \quad cf(1)=0$$

• (b) {f ∈ V | f(0) = 1}

$$\vec 0\not\in V$$ $$f_1(0)+f_2(0)=2$$

• (c) {f ∈ V | f only has a finite number of zeros} $$\vec0\not\in V$$ $$f_1(x)=0 \iff x=0 \quad f_2(x)=0 \iff x=1$$ $$f_1+f_2\neq0$$

• (d) {f ∈ V | f has infinitely many zeros}

$$\vec0\in V$$

but refer to qbert example.

Answer 3

For problem d, my hint would be to examine the following functions $$ f_1(x)=\begin{cases}0 &x>0\\1 &x\leq0 \end{cases} $$ and $$ f_2(x)=\begin{cases}1 &x>0\\0 &x\leq0 \end{cases} $$ specifically, their sum.