A question about finite dimensional function vector space. Let $V$ be a nontrivial finite-dimensional subspace of continuous functions on $[0,1]$, prove that there exists $g \in V$, such that $\int^1_0g^2=1$ and $\max g^2\ge\dim V$.
(Assuming we're talking about real-valued functions:)

Say $f_1,\dots, f_n$ is an orthonormal basis for $V\subset L^2([0,1])$. Since $$\int_0^1\sum_{j=1}^n f_j^2=n$$there must exist $t\in[0,1]$ with $$\sum_{j=1}^n f_j(t)^2\ge n.$$Hence there exist scalars $a_1,\dots a_n$ with $\sum a_j^2=1$ and $$\sum a_jf_j(t)\ge\sqrt n.$$
(Assuming we're talking about real-valued functions:)

Say $f_1,\dots, f_n$ is an orthonormal basis for $V\subset L^2([0,1])$. Since $$\int_0^1\sum_{j=1}^n f_j^2=n$$there must exist $t\in[0,1]$ with $$\sum_{j=1}^n f_j(t)^2\ge n.$$Hence there exist scalars $a_1,\dots a_n$ with $\sum a_j^2=1$ and $$\sum a_jf_j(t)\ge\sqrt n.$$