A Banach algebra with an unbounded approximate identity.

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Consider l^{p} as a Banach algebra with coordinatewise operations. Let e_{n} =

(\underbrace{1, 1, 1, …, 1}_{n}, 0, 0, …). Then sup_{n} ||e_{n}|| = sup\{ ^{p} \sqrt{n} ; n ∈ N\} = ∞, and for every

x = (α_{n}) ∈ l^{2}, lim_{n} ||xe_{n} − x|| = lim_{n}(∑_{k=n+1}^{∞} |α_{k}|^{p})\frac{1}{p} = 0. Thus (e_{n}) is required

approximate identity.