Linear Algebra - Proof of subspace

Question

I'm stuck in how to proof the follow question.

Let $V = \mathcal{F}({I},\mathbb{R}) $, where ${I}$ is in the range of $[0,1]$ and $$T = \left\{ \mathcal{f} \in V: f(1) = 0 \right\} $$
Is $T$ a vectorial subspace of V?

How can I demonstrate that?

Answer 1

Take any $f,g\in V$, then $f(1)=g(1)=0$ so $$(f+g)(1) = f(1)+g(1)=0+0=0$$ so $f+g \in V$ and similary for multipliction with scalar $$(a\cdot f)(1) = af(1)=a\cdot 0=0$$

so $a\cdot f\in V$ also.

Answer 2

first I think they must be a mistake in your set, and you mean $ {T={f∈V:f(1)=0} } $. Then you have to ask yourself about axioms of vectorial space. Here is a little question: if $ f $ and $ g \in V $ does $ f + g $ belong to V ?